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

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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.
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 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.
(e^x – 1)² e^-4x
Solution:

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

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