RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Textbook Exercise Questions and Answers.

RBSE Class 12 Maths Solutions Chapter 7 Integrals Ex 7.4

Question 1.
\(\frac{3 x^{2}}{x^{6}+1}\)
Answer:

Question 2.
\(\frac{1}{\sqrt{1+4 x^{2}}}\)
Answer:

Question 3.
\(\frac{1}{\sqrt{(2-x)^{2}+1}}\)
Answer:

Question 4.
\(\frac{1}{\sqrt{9-25 x^{2}}}\)
Answer:

Question 5.
\(\frac{3 x}{1+2 x^{4}}\)
Answer:

Question 6.
\(\frac{x^{2}}{1-x^{6}}\)
Answer:

Question 7.
\(\frac{x-1}{\sqrt{x^{2}-1}}\)
Answer:

Question 8.
\(\frac{x^{2}}{\sqrt{x^{6}+a^{6}}}\)
Answer:

Question 9.
\(\frac{\sec ^{2} x}{\sqrt{\tan ^{2} x+4}}\)
Answer:
Let I = ∫\(\frac{\sec ^{2} x}{\sqrt{\tan ^{2} x+4}}\) dx
Putting tan x = t
⇒ sec² x dx = dt
∴ I = ∫\(\frac{d t}{\sqrt{t^{2}+4}}\)
= log |t + \(\sqrt{t^{2}+4}\)| + C
= log |tan x + \(\sqrt{\tan ^{2} x+4}\)| + C

Question 10.
\(\frac{1}{\sqrt{x^{2}+2 x+2}}\)
Answer:

Question 11.
\(\frac{1}{9 x^{2}+6 x+5}\)
Answer:

Question 12.
\(\frac{1}{\sqrt{7-6 x-x^{2}}}\)
Answer:

Question 13.
\(\frac{1}{\sqrt{(x-1)(x-2)}}\)
Answer:

Question 14.
\(\frac{1}{\sqrt{8+3 x-x^{2}}}\)
Answer:

Question 15.
\(\frac{1}{\sqrt{(x-a)(x-b)}}\)
Answer:

Question 16.
\(\frac{4 x+1}{\sqrt{2 x^{2}+x-3}}\)
Answer:

Question 17.
\(\frac{x+2}{\sqrt{x^{2}-1}}\)
Answer:

Question 18.
\(\frac{5 x-2}{1+2 x+3 x^{2}}\)
Answer:
Let I = ∫\(\frac{5 x-2}{1+2 x+3 x^{2}}\) dx
Let A and B are two numbers such that:
5x - 2 = A \(\frac{d}{d x}\) (1 + 2x + 3x²) + B
⇒ 5x - 2 = A(2 + 6x) + B
⇒ 5x - 2 = 6Ax + 2A + B
Comparing the coefficients of x and constant terms in both sides,

Question 19.
\(\frac{6 x+7}{\sqrt{(x-5)(x-4)}}\)
Answer:
Let I = ∫\(\frac{6 x+7}{\sqrt{(x-5)(x-4)}}\) dx
= ∫\(\frac{6 x+7}{\sqrt{(x-5)(x-4)}}\) dx
Let A and B are two numbers such that:
6x + 7 = A \(\frac{d}{d x}\)(x² - 9x + 20) + B
⇒ 6x + 7 = A(2x - 9) + B
⇒ 6x + 7 = 2A - 9A + B
Comparing the coefficients of x and constant terms in both sides, we get
2A = 6 and - 9A + B = 7
A = 3 and - 27 + B = 7 ⇒ B = 27 + 7 = 34

Question 20.
\(\frac{x+2}{\sqrt{4 x-x^{2}}}\)
Answer:

Question 21.
\(\frac{x+2}{\sqrt{x^{2}+2 x+3}}\)
Answer:

Question 22.
\(\frac{x+3}{x^{2}-2 x-5}\)
Answer:

Question 23.
∫\(\frac{5 x+3}{\sqrt{x^{2}+4 x+10}}\)
Answer:
Let I = ∫\(\frac{5 x+3}{\sqrt{x^{2}+4 x+10}}\) dx
Let A and B are two numbers such that:
5x + 3 = A \(\frac{d}{d x}\) (x² + 4x + 10) + B
⇒ 5x + 3 = A(2x + 4) + B
⇒ 5x + 3 = 2Ax + 4A + B
Comparing the coefficients of x and constant terms from both sides, we have

Question 24.
∫\(\frac{d x}{x^{2}+2 x+2}\) equals:
(A) x tan^-1 (x + 1) + C
(B) tan^-1 (x + 1) + C
(C) (x + 1) sin^-1 x + C
(D) tan^-1 x + C
Answer:
∫\(\frac{d x}{x^{2}+2 x+2}\) = ∫\(\frac{d x}{x^{2}+2 x+1+1}\)
= ∫\(\frac{d x}{(x+1)^{2}+1^{2}}\) = tan^-1 \(\left(\frac{x+1}{1}\right)\) + C
= tan^-1 (x + 1) + C
Hence, (B) is the correct answer.

Question 25.
∫\(\frac{d x}{\sqrt{9 x-4 x^{2}}}\) equals:
(A) \(\frac{1}{9} \sin ^{-1}\left(\frac{9 x-8}{8}\right)+C\)
(B) \(\frac{1}{2} \sin ^{-1}\left(\frac{8 x-9}{9}\right)+C\)
(C) \(\frac{1}{2} \sin ^{-1}\left(\frac{9 x-8}{9}\right)+C\)
(D) \(\frac{1}{3} \sin ^{-1}\left(\frac{9 x-8}{9}\right)+C\)
Answer:

Hence, (B) is the correct answer.