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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5

Integrate the following functions with respect to x
Question 1.
\(\frac{x^{3}+4 x^{2}-3 x+2}{x^{2}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 1

Question 2.
\(\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{2}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 2

Question 3.
(2x – 5)(36 + 4x)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 3

Question 4.
cot2 x + tan2 x
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 4

Question 5.
\(\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 5

Question 6.
\(\frac{\cos 2 x}{\sin ^{2} x \cos ^{2} x}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 6

Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5

Question 7.
\(\frac{3+4 \cos x}{\sin ^{2} x}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 7

Question 8.
\(\frac{\sin ^{2} x}{1+\cos x}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 8

Question 9.
\(\frac{\sin 4 x}{\sin x}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 9

Question 10.
cos 3x cos 2x
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 10

Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5

Question 11.
sin2 5x
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 11

Question 12.
\(\frac{1+\cos 4 x}{\cot x-\tan x}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 12
Question 13.
ex log aex
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 13

Question 14.
(3x + 4) \(\sqrt{3 x+7}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 14
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 15

Question 15.
\(\frac{8^{1+x}+4^{1-x}}{2^{x}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 16

Question 16.
\(\frac{1}{\sqrt{x+3}-\sqrt{x-4}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 17

Question 17.
\(\frac{x+1}{(x+2)(x+3)}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 18
x + 1 = A(x + 3) + B(x + 2) ………….. (i)
Put x = -3 in (i)
-2 = B(-1)
B = 2
Put x = -2 in (i)
-1 = A(1)
A = -1
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 19

Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5

Question 18.
\(\frac{1}{(x-1)(x+2)^{2}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 20

Question 19.
\(\frac{3 x-9}{(x-1)(x+2)\left(x^{2}+1\right)}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 21
put x = 1 in (i)
3 – 9 = A(3)(2) + 0 + 0
– 6 = 6A
A = -1

put x = -2 in (i)
-6 – 9 = 0 + B(-3) (5) + 0
-15 = – 15B
B = 1

put x = 0 in (i)
0 – 9 = A(2)(1) + B(-1)(1) + D(-1)(2)
-9 = 2A – B – 2D
2D = 2A – B + 9
2D = 2(-1) – 1 + 9
= -2 – 1 + 9
2D = 6
D = 3

Equating x3 coefficients on b/s in (i)
A + B + C = 0
-1 + 1 + C = 0
C = 0
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 22

Question 20.
\(\frac{x^{3}}{(x-1)(x-2)}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 23
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 24

Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 Additional Problems

Integrate the following
Question 1.
\(\frac{e^{2 x}+e^{-2 x}+2}{e^{x}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 25

Question 2.
cos3 2x – sin 6x
Solution:
cos 3x = 4 cos3 x – 3 cos x
∴ 4 cos3 x = cos 3x + 3 cos x
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 26

Question 3.
\(\sqrt{1-\sin 2 x}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 27

Question 4.
cos 2x sin 4x
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 28
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 29

Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5

Question 5.
(ex – 1)2 e-4x
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 30

Question 6.
(x + 1) \(\sqrt{x+3}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 31

Question 7.
(2x + 1)\(\sqrt{2x+3}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 32
Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5 33

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