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

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.