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

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}+1right)^{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}+9right)})
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}+1right)left(x^{2}+4right)})
Solution:

Question 15.
cos³xe^{log sinx}
Solution:

Question 16.
e^3logx(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})e^x
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}+1right)-2 log xright]}{x^{4}})
Solution:

Evaluate the following definite integrals from questions 25 to 33:
Question 25.
(int_{frac{pi}{2}}^{π}) e^x((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.
(intfrac{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 = t² ⇒sin 2x = 1 – t^2.
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}) xe^x dx = 1
Solution:
Let L.H.S. = I = (int_{0}^{1}) xe^x 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})e^2-3x dx as a limit of a sum.
Solution:
Here, a = 0, b = 1, f(x) = e^2-3x, 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(e^x – e^-x) + C
(D) log(e^x + 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.