These NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.8 Questions and Answers are prepared by our highly skilled subject experts.

## NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.8

Question 1.

\(\int_{a}^{b} x d x\)

Solution:

Let I = \(\int_{a}^{b} x d x\)

f(x) = x, nh = b – a

f(a) = a

f[a + h) = a + h

f(a + 2 h) = a+ 2h,

……………………….

f[a + (n – 1)h) = a + (n – 1)h

Question 2.

\(\int_{0}^{5}(x+1) d x\)

Solution:

Let I = \(\int_{0}^{5}(x+1) d x\)

We have a = 0, b = 5 and f(x) = x + 1

nh = b-a = 5 – 0 = 5

f(0) = 0 + 1 = 1

f(0 + h) = 0 + h + 1 = h + 1

f(0 + 2h) = 0 + 2h + 1 = 2h + 1

………………………….

f(0 + (n – 1 )h) = 0 + (n – 1)h + 1 = (n – 1)h + 1

Question 3.

\(\int_{2}^{3} x^{2} d x\)

Solution:

Let I = \(\int_{2}^{3} x^{2} d x\)

f(x) = x², a = 2, b = 3, nh = b – a = 3 – 2 = 1

f(2) = 2² = 4,

f(2 + h) = (2 + h)² = 4 + h² + 4h

f(2 + 2h) = (2 + 2h)² = 4+ 4h² + 8h

………………………….

f(2 + (n – 1 )h) = (2 + (n – 1)h]² = 4 + (h – 1)²h² + 4(n – 1)h

Question 4.

\(\int_{1}^{4}\left(x^{2}-x\right) d x\)

Solution:

Let I = \(\int_{1}^{4}\left(x^{2}-x\right) d x\)

We have a = 1, b = 4, f(x) = x² – x and nh = b – a = 4 – 1 = 3

f(1) = 1² – 1 = 0

f(1 + h) = (1 + h)² – (1 + h) = h² + h

f(1 + 2h) = (1 + 2h)² – (1 + 2 h) = 2²h² + 2 h

………………………….

f(1 + n – 1)h) = [1 + (n – 1)h]² – [1 + (n – 1)h] = (n – 1)² h² +(n – 1 )h

Question 5.

\(\int_{-1}^{1} e^{x} d x\)

Solution:

Let I = \(\int_{-1}^{1} e^{x} d x\)

We have a = – 1, b = 1, f(x) = e^{x}, nh = b – a = 1 + 1 = 2

f(- 1) = e^{-1}

f(- 1 + h) = e^{-1+h}

f(- 1 + 2h) = e^{-1+2h}

………………………….

f(- 1 + (n – h)h) = e^{-1+(n-1)h}

Question 6.

\(\int_{0}^{4}\left(x+e^{2 x}\right) d x\)

Solution:

Let I = \(\int_{0}^{4}\left(x+e^{2 x}\right) d x\)

f(x) = x + e^{2x}

a = 0, b = 4, nh = b – a = 4 – 0 = 4

f(0) = 0 + e^{0} = 1

f(0 + h) = (0 + h) + e^{2(0+h)} = h + e^{2h}

f(0 + 2h) = (0 + 2h) + e^{2(0+h)} = 2h + e^{2h}

………………………….

f(0 + (n – 1)h) = (0 + (n – 1 )h) + e^{2(0+h)}=(n – 1 )h + e^{2(n-1)h}