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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.

cot^{2} x + tan^{2} 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.

sin^{2} 5x

Solution:

Question 12.

\(\frac{1+\cos 4 x}{\cot x-\tan x}\)

Solution:

Question 13.

e^{x log a}e^{x}

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 x^{3} 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.

cos^{3} 2x – sin 6x

Solution:

cos 3x = 4 cos^{3} x – 3 cos x

∴ 4 cos^{3} x = cos 3x + 3 cos x

Question 3.

\(\sqrt{1-\sin 2 x}\)

Solution:

Question 4.

cos 2x sin 4x

Solution:

Question 5.

(e^{x} – 1)^{2} e^{-4x}

Solution:

Question 6.

(x + 1) \(\sqrt{x+3}\)

Solution:

Question 7.

(2x + 1)\(\sqrt{2x+3}\)

Solution: