# GSEB Solutions Class 12 Maths Chapter 7 Integrals Miscellaneous Exercise

Gujarat Board GSEB Textbook Solutions Class 12 Maths Chapter 7 Integrals Miscellaneous Exercise Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 11 Maths Chapter 7 Integrals Miscellaneous Exercise

Integrate the following functions w.r.t x (questions from 1 to 24):
Question 1.
$$\frac{1}{x-x^{3}}$$
Solution:
Let $$\frac{1}{x-x^{3}}$$ = $$\frac{1}{x(1+x)(1-x)}$$
= $$\frac{A}{x}$$ + $$\frac{B}{1+x}$$ + $$\frac{C}{1-x}$$
⇒ 1 = A(1 + x)(1 – x) + Bx(1 – x) + Cx(1 + x) ………… (1)
Putting x = 0 in (1), we get
1 = A(1 + 0)(1 – 0) ⇒ A = 1.
Putting x = – 1 in (1), we get
1 = B(- 1)(1 + 1) ⇒ B = – $$\frac{1}{2}$$
Putting x = 1 in (1), we get
1 = C(1)(1 + 1) ⇒ C = $$\frac{1}{2}$$.

Question 2.
$$\frac{1}{\sqrt{x+a}+\sqrt{x+b}}$$
Solution:

Question 3.
$$\frac{1}{x \sqrt{a x-x^{2}}}$$
Solution:

Question 4.
$$\frac{1}{x^{2}\left(x^{4}+1\right)^{\frac{3}{4}}}$$
Solution:

Question 5.
$$\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$$
Solution:

Question 6.
$$\frac{5 x}{(x+1)\left(x^{2}+9\right)}$$
Solution:

Question 7.
$$\frac{sinx}{sin(x-a)}$$
Solution:

= cos a∫1 dx + sin a∫cot(x – a) dx
= (cos a)x + sin a log |sin(x – a)| + C
= x cos a + sin a log + |sin(x – a)| + C.

Question 8.
$$\frac{e^{5 \log x}-e^{4 \log x}}{e^{3 \log x}-e^{2 \log x}}$$
Solution:

Question 9.
$$\frac{\cos x}{\sqrt{4-\sin ^{2} x}}$$
Solution:

Question 10.
$$\frac{\sin ^{8} x-\cos ^{8} x}{1-2 \sin ^{2} x \cos ^{2} x}$$
Solution:

Question 11.
$$\frac{1}{cos(x+a)cos(x+b)}$$
Solution:

Question 12.
$$\frac{x^{3}}{\sqrt{1-x^{8}}}$$
Solution:

Question 13.
$$\frac{e^{x}}{\left(1+e^{x}\right)\left(2+e^{x}\right)}$$
Solution:

Question 14.
$$\frac{1}{\left(x^{2}+1\right)\left(x^{2}+4\right)}$$
Solution:

Question 15.
cos3xelog sinx
Solution:

Question 16.
e3logx(x4 + 1)-1
Solution:

Question 17.
f'(ax + b)[f(ax + b)]n
Solution:

Question 18.
$$\frac{1}{\sqrt{\sin ^{3} x \sin (x+\alpha)}}$$
Solution:

Put cos α + cot x sin α = t so that – cosec2x sin α = dt.

Question 19.
$$\frac{\sin ^{-1} \sqrt{x}-\cos ^{-1} \sqrt{x}}{\sin ^{-1} \sqrt{x}+\cos ^{-1} \sqrt{x}}$$ x ∈ [0, 1]
Solution:

Question 20.
$$\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$$
Solution:

Question 21.
$$\frac{2+sin2x}{1+cos2x}$$ex
Solution:

Question 22.
$$\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}$$
Solution:

Question 23.
tan-1$$\sqrt{\frac{1-x}{1+x}}$$
Solution:
Put x = cosθ so that dx = – sinθ dθ,

Question 24.
$$\frac{\sqrt{x^{2}+1}\left[\log \left(x^{2}+1\right)-2 \log x\right]}{x^{4}}$$
Solution:

Evaluate the following definite integrals from questions 25 to 33:
Question 25.
$$\int_{\frac{\pi}{2}}^{π}$$ ex($$\frac{1-sinx}{1+cosx}$$)dx
Solution:

Question 26.
$$\int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos ^{4} x+\sin ^{4} x}$$ dx
Solution:

Question 27.
$$\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{2} x}{\cos ^{2} x+4 \sin ^{2} x}$$ dx
Solution:

Question 28.
$$\int\frac{3}{6} \frac{\sin x+\cos x}{\sqrt{\sin 2 x}}$$ dx
Solution:

Question 29.
$$\int_{0}^{1} \frac{d x}{\sqrt{1+x}-\sqrt{x}}$$
Solution:

Question 30.
$$\int_{0}^{\frac{\pi}{4}} \frac{\sin x+\cos x}{9+16 \sin 2 x}$$ dx
Solution:
Put sin x – cos x = t so that (cos x + sin x) dx = dt
and 1 – 2sin x cos x = t ⇒sin 2x = 1 – t2.
When x = $$\frac{π}{4}$$, t = sin $$\frac{π}{4}$$ – cos $$\frac{π}{4}$$ = $$\frac{1}{\sqrt{2}}$$ – $$\frac{1}{\sqrt{2}}$$ = 0.
When x = 0, t = sin 0 – cos 0 = – 1.

Question 31.
$$\int_{0}^{\frac{\pi}{2}}$$sin2xtan-1x(sin x)dx
Solution:

Question 32.
$$\int_{0}^{\pi}$$ $$\frac{xtanx}{secx+tanx}$$ dx
Solution:

Question 33.
$$\int_{1}^{4}$$[|x – 1| + |x – 2| + |x – 3|] dx
Solution:

Prove the following questions 34 to 39:
Question 34.
$$\int_{1}^{3} \frac{d x}{x^{2}(x+1)}$$ = $$\frac{2}{3}$$ + log $$\frac{2}{3}$$
Solution:

Question 35.
$$\int_{0}^{1}$$ xex dx = 1
Solution:
Let L.H.S. = I = $$\int_{0}^{1}$$ xex dx.
Integrating by parts, taking x as a first function, we get

Question 36.
$$\int_{-1}^{-1}$$ x17cos4x dx = 0
Solution:
I = $$\int_{-1}^{1}$$ x17cos4x dx.
Let f(x) = x17cos4x, f(- x) = (- x)17cos4(- x)
= – x17cos4x
∴ I = 0 = R.H.S. [∵ $$\int_{-a}^{a}$$ f(x) = 0 if f(- x) = – f(x)]

Question 37.
$$\int_{0}^{\frac{\pi}{2}}$$sin3x dx = $$\frac{2}{3}$$
Solution:

Question 38.
$$\int_{0}^{\frac{\pi}{2}}$$2tan3 x dx = 1 – log x
Solution:

Question 39.
$$\int_{0}^{1}$$sin-1x dx = $$\frac{π}{2}$$ – 1
Solution:

Question 40.
Evaluate $$\int_{0}^{1}$$e2-3x dx as a limit of a sum.
Solution:
Here, a = 0, b = 1, f(x) = e2-3x, h = $$\frac{1-0}{n}$$ = $$\frac{1}{n}$$ or nh = 1.

Choose the correct answers in the following questions 41 to 44:
Question 41.
∫$$\frac{d x}{e^{x}+e^{-x}}$$ is equal to
(A) tan-1(e-x) + C
(B) tan-1(e-x) + C
(C) log(ex – e-x) + C
(D) log(ex + e-x) + C
Solution:

∴ Part (A) is the correct answer.

Question 42.
∫$$\frac{\cos 2 x}{(\sin x+\cos x)^{2}}$$ dx is equal to
(A) $$\frac{-1}{sinx+cosx}$$ + C
(B) log|sin x + cos x| + C
(C) log|sin x – cos x| + C
(D) $$\frac{1}{(\sin x+\cos x)^{2}}$$
Solution:

∴ Part (B) is the correct answer.

Question 43.
If f(a + b – x) = f(x), then $$\int_{a}^{b}$$ x f(x) dx is equal to
(A) $$\frac{a+b}{2}$$ $$\int_{a}^{b}$$ f(b – x) dx
(B) $$\frac{a+b}{2}$$ $$\int_{a}^{b}$$ f(b + x) dx
(C) $$\frac{b-a}{2}$$ $$\int_{a}^{b}$$ f(x) dx
(D) $$\frac{a+b}{2}$$ $$\int_{a}^{b}$$ f(x) dx
Solution:

∴ Part (D) is the correct answer.

Question 44.
The value of $$\int_{0}^{1}$$ tan-1($$\frac{2 x-1}{1+x-x^{2}}$$) dx is
(A) 1
(B) 0
(C) – 1
(D) $$\frac{π}{4}$$
Solution:

Adding (1) and (2), we get
2I = 0 or I = 0.
∴ Part (B) is the correct answer.