NCERT Board Solutions for Class 12 Maths Chapter 7 Integrals

प्रश्नावली 7.1

निम्नलिखित फलनों के प्रतिअवकलज ( समाकलन) निरीक्षण विधि द्वारा ज्ञात कीजिए।

प्रश्न 1.
sin 2x
हल-
∫sin 2x dx
हम जानते हैं कि, \frac { d }{ dx }

cos2x = -2 sin 2x
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1

प्रश्न 2.
cos 3x
हल-
∫cos 3x dx
हम जानते हैं कि, \frac { d }{ dx }sin 3x = 3 cos 3x
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.1

प्रश्न 3.
e2x
हल-
∫e2x dx
हम जानते हैं कि,
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3

प्रश्न 4.
(ax + b)²
हल-
∫(ax + b)² dx
हम जानते हैं कि,
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4

प्रश्न 5.
sin 2x – 4e3x
हल-
∫(sin 2x -4e3x) dx = ∫sin 2x dx – 4∫e3x …(1)
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5

निम्नलिखित समाकलनों को ज्ञात कीजिए।

प्रश्न 6.
∫(4e3x + 1) dx
हल-
∫(4e3x + 1) dx = 4∫e3xdx + ∫1 dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6

प्रश्न 7.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.1

प्रश्न 8.
∫(ax² + bx + c) dx
हल-
∫(ax² + bx + c) dx = a∫x²dx + b∫xdx + c∫1 dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8

प्रश्न 9.
∫(2x² + ex) dx
हल-
∫(2x² + ex) dx = 2∫x² dx + ∫ex dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9

प्रश्न 10.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.1

प्रश्न 11.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11.1

प्रश्न 12.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12.1

प्रश्न 13.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13.1

प्रश्न 14.
∫(1-x)√x dx
हल-
∫(1-x)√x dx
=∫(√x-x√x)dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14

प्रश्न 15.
∫√x(3x² + 2x + 3)dx
हल-
∫√x(3x² + 2x + 3)dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15

प्रश्न 16.
∫(2x – 3cosx + ex)dx
हल-
∫(2x – 3cosx + ex)dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16

प्रश्न 17.
∫(2x² – 3sinx + 5√x)dx
हल-
∫(2x² – 3sinx + 5√x)dx
= 2∫x² dx – 3 ∫ sinx dx + 5∫√x dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17.1

प्रश्न 18.
∫secx (sec x + tan x) dx
हल-
∫sec x (sec x + tan x) dx
= ∫(sec² x + sec x tan x) dx
= ∫(sec² x dx
= ∫sec x tan x dx
tan x + sec x + C

प्रश्न 19.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.1

प्रश्न 20.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.1

प्रश्न 21 एवं 22 में सही उत्तर का चयन कीजिए

प्रश्न 21
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21.1

प्रश्न 22
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22.2

प्रश्नावली 7.2

प्रश्न 1 से 37 तक के प्रश्नों में प्रत्येक फलने का समाकलन ज्ञात कीजिए

प्रश्न 1.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1.1

प्रश्न 2.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.1

प्रश्न 3.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3.1

प्रश्न 4.
sin x sin (cos x)
हल-
मानी I = ∫ sin x sin (cos x) dx …(1)
माना cos x = t, दोनों पक्षों का x के सापेक्ष अवकलन करने पर,
– sin x dx = dt ⇒ sin x dx = – dt
∴(1) से, I = ∫sin (cos x) (sin x) dx = -∫sint dt
= cos t + C = cos (cos x) + C

प्रश्न 5.
sin (ax + b) cos (ax + b)
हल-
माना I = ∫sin (ax + b) cos (ax + b) dx
माना sin (ax + b) = t, दोनों पक्षों का x के सापेक्ष अवकलन करने पर,
a cos (ax + b) dx = dt
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5

प्रश्न 6.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.1

प्रश्न 7.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.1

प्रश्न 8.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.2

प्रश्न 9.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9.1

प्रश्न 10.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.1

प्रश्न 11.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11.1

प्रश्न 12.
(x3 – 1)1/3 x5
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12.1

प्रश्न 13.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13.1

प्रश्न 14
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14.1

प्रश्न 15
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15.1

प्रश्न 16.
e2x+3
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16

प्रश्न 17.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17.1

प्रश्न 18.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.1

प्रश्न 19.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.1

प्रश्न 20.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.1

प्रश्न 21.
tan²(2x – 3)
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21

प्रश्न 22.
sec²(7 – 4x)
हल-
माना
I = ∫sec²(7 – 4x) …(1)
माना 7 – 4x = t, दोनों पक्षों का x के सापेक्ष अवकलन करने पर,
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22

प्रश्न 23.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23.1

प्रश्न 24.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 24
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 24.1

प्रश्न 25.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 25
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 25.1

प्रश्न 26.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 26
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 26.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 26.2

प्रश्न 27.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 27
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 27.1

प्रश्न 28.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 28
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 28.1

प्रश्न 29.
cot x log sin x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 29

प्रश्न 30.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 30
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 30.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 30.2

प्रश्न 31.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 31
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 31.1

प्रश्न 32.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 32
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 32.1

प्रश्न 33.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 33
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 33.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 33.2

प्रश्न 34.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 34
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 34.1

प्रश्न 35.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 35
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 35.1

प्रश्न 36.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 36
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 36.1

प्रश्न 37.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 37
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 37.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 37.2

प्रश्न 38.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 38
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 38.1

प्रश्न 39.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 39
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 39.1

प्रश्नावली 7.3

निर्देश- 1 से 22 तक के प्रश्नों में प्रत्येक का समाकलन ज्ञात कीजिए।

प्रश्न 1.
sin² (2x + 5)
हल-
माना I = ∫sin ² (2x + 5) dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1

प्रश्न 2.
sin 3x cos 4x dx
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2

प्रश्न 3.
cos 2x cos 4x cos 6x dx
हल-
cos 2x cos 4x cos 6x dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3

प्रश्न 4.
sin3 (2x + 1)
हल-
माना I = ∫sin3 (2x + 1) dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4.1

प्रश्न 5.
sin3x cos3x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5

प्रश्न 6.
sinx sin2x sin3x
हल-
माना I = ∫sinx sin2x sin3x
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6

प्रश्न 7.
sin 4x sin 8x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7

प्रश्न 8.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.1

प्रश्न 9.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9.1

प्रश्न 10.
sin xdx
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10

प्रश्न 11.
cos4 2x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11

प्रश्न 12.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12.1
हल-
माना
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12
= ∫(1-cosx)dx = ∫1 dx – ∫ cosx dx
= x – sinx +C

प्रश्न 13.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13.1

प्रश्न 14.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14.1

प्रश्न 15.
tan3 2x sec 2x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15

प्रश्न 16.
tan4 x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16

प्रश्न 17.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17.1

प्रश्न 18.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.1

प्रश्न 19.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.1

प्रश्न 20.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.2

प्रश्न 21.
sin-1 (cosx)
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21

प्रश्न 22.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22.1

प्रश्न 23.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23.1

प्रश्न 24.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 24
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 24.1

प्रश्नावली 7.4

प्रश्न 1 से 23 तक के फलनों का समाकलन कीजिए।

प्रश्न 1.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1.1

प्रश्न 2.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.1

प्रश्न 3.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3.1

प्रश्न 4.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4.1

प्रश्न 5.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5.1

प्रश्न 6.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.1

प्रश्न 7.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.2

प्रश्न 8.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.1

प्रश्न 9.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9.1

प्रश्न 10.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.1

प्रश्न 11.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11.1

प्रश्न 12.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12.1

प्रश्न 13.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13.2

प्रश्न 14.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14.1

प्रश्न 15.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15.1

प्रश्न 16.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16.1

प्रश्न 17.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17.1

प्रश्न 18.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.2
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.3

प्रश्न 19.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.2
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.3

प्रश्न 20.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.1

प्रश्न 21.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21.1

प्रश्न 22.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22.2

प्रश्न 23.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23.1

प्रश्न 24.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 24
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 24.1

प्रश्न 25.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 25
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 25.1

प्रश्नावली 7.5

प्रश्न 1 से 21 तक के प्रश्नों में परिमेय फलनों का समाकलन कीजिए

प्रश्न 1.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1.2

प्रश्न 2.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.1

प्रश्न 3.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3.2

प्रश्न 4.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4.1

प्रश्न 5.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5.1

प्रश्न 6.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.2

प्रश्न 7.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.2

प्रश्न 8.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.1

प्रश्न 9.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9.2

प्रश्न 10.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.2

प्रश्न 11.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11.1

प्रश्न 12.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12.2

प्रश्न 13.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13.1

प्रश्न 14.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14.1

प्रश्न 15.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15.1

प्रश्न 16.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16.1

प्रश्न 17.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17.1

प्रश्न 18.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.2

प्रश्न 19.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.1

प्रश्न 20.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.2

प्रश्न 21.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21.1

प्रश्न 22.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22.1

प्रश्न 23.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23.1

प्रश्नावली 7.6

प्रश्न 1 से 22 तक के प्रश्नों के फलनों का समाकलन कीजिए

प्रश्न 1.
फलन x sin x का x के सापेक्ष समाकल ज्ञात कीजिए।
हल-
∫x sin x dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1
= – x cos x + ∫cos x dx
= – x cos x + sin x + C

प्रश्न 2.
x sin 3x
हल-
माना I = ∫x.sin 3x dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2

प्रश्न 3.
x2 ex
हल-
माना I = ∫x2 ex dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3

प्रश्न 4.
फलन x log x का x के सापेक्ष समाकल ज्ञात कीजिए।
हल-
∫ x log x dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4

प्रश्न 5.
x log 2x
हल-
माना I = ∫x log 2x dx
= ∫(log 2x).x dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5

प्रश्न 6.
फलन x² log x का x के सापेक्ष समाकल ज्ञात कीजिए।
हल-
माना I = ∫x² log x dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6

प्रश्न 7.
x sin-1 x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7

प्रश्न 8.
x tan-1x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8

प्रश्न 9.
x cos-1 x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9

प्रश्न 10.
(sin-1x)2
हल-
माना I = [(sin-1 x)2 dx ….(1)
sin-1 x = θ ⇒ x = sinθ ⇒ dx = cos θ dθ
∴(1) से,
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10

प्रश्न 11.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11.1

प्रश्न 12.
x sec² x
हल-
माना I = ∫x sec² x dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12
x को पहला फलन तथा sec²x को दूसरा फलन लेकन खण्डश: समाकलन करने पर,
= x tan x – ∫1. tan x dx
= x tan x + log|cos x| + C

प्रश्न 13.
tan-1 x
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13

प्रश्न 14.
x(log x)²
हल-
माना I = ∫x (log x)² dx
= ∫(log x)².x dx
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14.1

प्रश्न 15.
(x² + 1) log x
हल-
माना I = ∫(x² + 1) log x dx
= ∫log x. (x² + 1) dx
(log x) को पहला फलन तथा (x² + 1) को दूसरा फलन लेकर खण्डश: समाकलन करने पर,
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15

प्रश्न 16.
e(sinx + cosx)
हल-
ex sinx = t
⇒ e(sinx + cosx)dx = dt
= ∫ex(sinx + cosx)dx = ∫dt = t + c
= ex sinx + c

प्रश्न 17
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17
हैल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17.2

प्रश्न 18.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.1

प्रश्न 19.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.1

प्रश्न 20.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.1

प्रश्न 21.
e2x sinx
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21

प्रश्न 22.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22.1

प्रश्न 23
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 23.1

प्रश्न 24.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 24
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 24.1

प्रश्नावली 7.7

प्रश्न 1 से 9 तक के फलनों का समाकलन कीजिए

प्रश्न 1.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1.1

प्रश्न 2.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.1

प्रश्न 3.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3.1

प्रश्न 4.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4.1

प्रश्न 5.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5.1

प्रश्न 6.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.1

प्रश्न 7.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.2

प्रश्न 8.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.1

प्रश्न 9.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9.1

प्रश्न 10.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.1

प्रश्न 11.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11.1

प्रश्नावली 7.8

योगों की सीमा के रूप में निम्नलिखित निश्चित समाकलनों का मान ज्ञात कीजिए।

प्रश्न 1.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1.1

प्रश्न 2.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.1

प्रश्न 3.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3.1

प्रश्न 4.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4.1

प्रश्न 5.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5.1

प्रश्न 6.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.1

प्रश्नावली 7.9

1 से 20 तक के प्रश्नों में निश्चित समाकलनों का मान ज्ञात कीजिए

प्रश्न 1.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1.1

प्रश्न 2.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.1

प्रश्न 3.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3.1

प्रश्न 4.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4.1

प्रश्न 5.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5.1

प्रश्न 6.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.1

प्रश्न 7.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.1

प्रश्न 8.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.2

प्रश्न 9.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9.1

प्रश्न 10.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.1

प्रश्न 11.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11.1

प्रश्न 12.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12.1

प्रश्न 13.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13.2

प्रश्न 14.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14.1

प्रश्न 15.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15.2

प्रश्न 16.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16.2
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16.3

प्रश्न 17.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17.1

प्रश्न 18.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.1

प्रश्न 19.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.1

प्रश्न 20.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.1

प्रश्न 21.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 21.1

प्रश्न 22.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 22.1

प्रश्नावली 7.10

प्रश्न 1 से 8 तक के प्रश्नों में समाकलनों का मान प्रतिस्थापन का उपयोग करते हुए ज्ञात कीजिए।

प्रश्न 1.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1.1

प्रश्न 2.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.2

प्रश्न 3.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3.1

प्रश्न 4.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4.1

प्रश्न 5.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5.2

प्रश्न 6.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.2

प्रश्न 7.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.1

प्रश्न 8.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.1

प्रश्न 9.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9.1

प्रश्न 10.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.1

प्रश्नावली 7.11.

निश्चित समाकलनों में गुणधर्मों का उपयोग करते हुए 1 से 19 तक के प्रश्नों में समाकलनों का मान ज्ञात कीपिए—

प्रश्न 1.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 1.1

प्रश्न 2.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 2.1

प्रश्न 3.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 3.1

प्रश्न 4.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 4.1

प्रश्न 5.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 5.1

प्रश्न 6.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 6.1

प्रश्न 7.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 7.1

प्रश्न 8.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 8.1

प्रश्न 9.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 9.1

प्रश्न 10.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 10.2

प्रश्न 11.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 11.1

प्रश्न 12.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 12.1

प्रश्न 13.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 13.1

प्रश्न 14.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 14.1

प्रश्न 15.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 15.1

प्रश्न 16.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16.2
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 16.3

प्रश्न 17.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 17.1

प्रश्न 18.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.1
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 18.2

प्रश्न 19.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 19.1

प्रश्न 20.
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20
हल-
UP Board Solutions for Class 12 Maths Chapter 7 Integrals 20.1

प्रश्न 21.

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