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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:

Question 2.
$$\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{2}$$
Solution:

Question 3.
(2x – 5)(36 + 4x)
Solution:

Question 4.
cot2 x + tan2 x
Solution:

Question 5.
$$\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}$$
Solution:

Question 6.
$$\frac{\cos 2 x}{\sin ^{2} x \cos ^{2} x}$$
Solution:

Question 7.
$$\frac{3+4 \cos x}{\sin ^{2} x}$$
Solution:

Question 8.
$$\frac{\sin ^{2} x}{1+\cos x}$$
Solution:

Question 9.
$$\frac{\sin 4 x}{\sin x}$$
Solution:

Question 10.
cos 3x cos 2x
Solution:

Question 11.
sin2 5x
Solution:

Question 12.
$$\frac{1+\cos 4 x}{\cot x-\tan x}$$
Solution:

Question 13.
ex log aex
Solution:

Question 14.
(3x + 4) $$\sqrt{3 x+7}$$
Solution:

Question 15.
$$\frac{8^{1+x}+4^{1-x}}{2^{x}}$$
Solution:

Question 16.
$$\frac{1}{\sqrt{x+3}-\sqrt{x-4}}$$
Solution:

Question 17.
$$\frac{x+1}{(x+2)(x+3)}$$
Solution:

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

Question 18.
$$\frac{1}{(x-1)(x+2)^{2}}$$
Solution:

Question 19.
$$\frac{3 x-9}{(x-1)(x+2)\left(x^{2}+1\right)}$$
Solution:

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

Question 20.
$$\frac{x^{3}}{(x-1)(x-2)}$$
Solution:

### 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:

Question 2.
cos3 2x – sin 6x
Solution:
cos 3x = 4 cos3 x – 3 cos x
∴ 4 cos3 x = cos 3x + 3 cos x

Question 3.
$$\sqrt{1-\sin 2 x}$$
Solution:

Question 4.
cos 2x sin 4x
Solution:

Question 5.
(ex – 1)2 e-4x
Solution:

Question 6.
(x + 1) $$\sqrt{x+3}$$
Solution:

Question 7.
(2x + 1)$$\sqrt{2x+3}$$
Solution: