Evaluate the following definite integral as limit of sums.

Ex 7.8 Class 12 Maths Question 1.
$\int _{ a }^{ b }{ x\quad dx }$

Solution:
on comparing
$\int _{ a }^{ b }{ x\quad dx } \quad with\quad \int _{ a }^{ b }{ f(x)dx }$
we

have

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.8 Q1.1

Ex 7.8 Class 12 Maths Question 2.
$\int _{ 0 }^{ 5 }{ (x+1)dx }$
Solution:
on comparing
$\int _{ 0 }^{ 5 }{ (x+1)dx } \quad with\quad \int _{ 0 }^{ 5 }{ f(x)dx }$
we have f(x) = x+1, a = 0, b = 5
and nh = b-a = 5-0 = 5
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.8 Q2.1

Ex 7.8 Class 12 Maths Question 3.
$\int _{ 2 }^{ 3 }{ { x }^{ 2 } } dx$
Solution:
compare
$\int _{ 2 }^{ 3 }{ { x }^{ 2 } } dx\quad with\quad \int _{ a }^{ b }{ f({ x }) } dx$
we have
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.8 Q3.1

Ex 7.8 Class 12 Maths Question 4.
$\int _{ 1 }^{ 4 }{ ({ x }^{ 2 }-x) } dx$
Solution:
compare
$\int _{ 1 }^{ 4 }{ ({ x }^{ 2 }-x) } dx\quad with\quad \int _{ a }^{ b }{ f({ x }) } dx$
we have f(x) = x²-x and a = 1, b = 4
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.8 Q4.1

Ex 7.8 Class 12 Maths Question 5.
$\int _{ -1 }^{ 1 }{ { e }^{ x } } dx\quad$
Solution:
compare
$\int _{ -1 }^{ 1 }{ { e }^{ x } } dx\quad with\quad \int _{ a }^{ b }{ f({ x }) } dx$
we have
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.8 Q5.1

Ex 7.8 Class 12 Maths Question 6.
$\int _{ 0 }^{ 4 }{ { (x+e }^{ 2x }) } dx\quad$
Solution:
let f(x) = x + e^2x,
a = 0, b = 4
and nh = b – a = 4 – 0 = 4

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