GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.2

Gujarat Board GSEB Textbook Solutions Class 12 Maths Chapter 7 Integrals Ex 7.2 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 11 Maths Chapter 7 Integrals Ex 7.2

Integrate the following:
Question 1.
\(\frac{2 x}{1+x^{2}}\)
Solution:

Question 2.
\(\frac{(\log x)^{2}}{x}\)
Solution:

Question 3.
\(\frac{1}{x+xlogx}\)
Solution:

I = ∫\(\frac{1}{t}\)dt = log t + C
= log |1 + log x| + C.

Question 4.
sinx sin(cos x)
Solution:
Let I = ∫sin x sin(cos x) dx. Put cos x = t
∴ I = – ∫sin(cos x).(- sin x)dx
= – ∫sint dt
= cos t + C = cos(cos x) + C.

Question 5.
sin(ax + b)cos(ax + b)
Solution:
Let I = ∫sin(ax + b)cos(ax + b)dx
= \(\frac{1}{2}\) ∫2 sin(ax + b) cos(ax + b)dx
= \(\frac{1}{2}\) ∫sin(2ax + 2b)dx.
Put 2ax + 2b = t so that 2a dx = dt.

Question 6.
\(\sqrt{ax+b}\)
Solution:
Let I = ∫\(\sqrt{ax+b}\)dx = ∫(ax + b)^1/2dx
Put ax + b = t so that a dx = dt

Question 7.
x\(\sqrt{x+2}\)
Solution:
Let I = ∫x\(\sqrt{x+2}\)dx = ∫[(x + 2) – 2]\(\sqrt{x+2}\)dx

Question 8.
x\(\sqrt{1+2 x^{2}}\)
Solution:
Let I = ∫x\(\sqrt{1+2 x^{2}}\)dx.
Put 1 + 2x² = t so that 4x dx = dt.

Question 9.
(4x + 2)\(\sqrt{x^{2}+x+1}\)
Solution:
Let I = ∫(4x + 2)\(\sqrt{x^{2}+x+1}\)dx = 2∫\(\sqrt{x^{2}+x+1}\)(2x + 1)dx.
Put x² + x + 1 = t so that (2x + 1)dx = dt.

Question 10.
\(\frac{1}{x-\sqrt{x}}\)
Solution:

Question 11.
\(\frac{x}{\sqrt{x+4}}\)
Solution:

Question 12.
(x³ – 1)^1/3.x⁵
Solution:

Question 13.
\(\frac{x^{2}}{\left(2+3 x^{3}\right)^{3}}\)
Solution:

Question 14.
\(\frac{1}{x(\log x)^{m}}\)
Solution:

Question 15.
\(\frac{x}{9-4 x^{2}}\)
Solution:

Question 16.
e^2x+3
Solution:

Question 17.
\(\frac{x}{e^{x^{2}}}\)
Solution:

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

Quesrtion 19.
\(\frac{e^{2 x}-1}{e^{2 x}+1}\)
Solution:

Question 20.
\(\frac{e^{2 x}-e^{-2 x}}{e^{2 x}+e^{+2 x}}\)
Solution:

Question 21.
tan²(2x – 3)
Solution:

Question 22.
sec²(7 – 4x)
Solution:

Question 23.
\(\frac{\sin ^{-1} x}{\sqrt{1-x^{2}}}\)
Solution:

Question 24.
\(\frac{2cosx-3sinx}{6cosx+4sinx}\)
Solution:

Question 25.
\(\frac{1}{\cos ^{2} x(1-\tan x)^{2}}\)
Solution:

Question 26.
\(\frac{\cos \sqrt{x}}{\sqrt{x}}\)
Solution:

Question 27.
\(\sqrt{sin2x}\) cos2x
Solution:

Question 28.
\(\frac{\cos x}{\sqrt{(1+\sin x)}}\)
Solution:

Question 29.
cot x log sin x
Solution:

Question 30.
\(\frac{sinx}{1+cosx}\)
Solution:

Question 31.
\(\frac{\sin x}{(1+\cos x)^{2}}\)
Solution:

Question 32.
\(\frac{1}{1+cotx}\)
Solution:

Question 33.
\(\frac{1}{1-tanx}\)
Solution:

Question 34.
\(\frac{\sqrt{\tan x}}{\sin x \cos x}\)
Solution:

Question 35.
\(\frac{(1+\log x)^{2}}{x}\)
Solution:

Question 36.
\(\frac{(x+1)(x+\log x)^{2}}{x}\)
Solution:

Question 37.
\(\frac{x^{3} \sin \left(\tan ^{-1} x^{4}\right)}{1+x^{8}}\)
Solution:

Choose the correct answers in the following questions 38 and 39:
38. ∫\(\frac{10 x^{9}+10^{x} \log _{e} 10}{x^{10}+10^{x}}\)dx equals
(A) 10^x – x¹⁰ + C
(B) 10^x + x¹⁰ + C
(C) (10^x – x¹⁰)^-1 + C
(D) log(10^x + x¹⁰ + C)
Solution:

∴ Part(D) is the correct answer.

Question 39.
∫\(\frac{d x}{\sin ^{2} x \cos ^{2} x}\) equals to
(A) tan x + cot x + C
(B) tan x – cot x + C
(C) tan x cot x + C
(D) tan x cot x + C
Solution:

∴ Part(B) is the correct answer.