Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 3 Indefinite Integration Ex 3.3 Questions and Answers.

## Maharashtra State Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

I. Evaluate the following:

Question 1.

∫x^{2} log x dx

Solution:

Question 2.

∫x^{2} sin 3x dx

Solution:

Question 3.

∫x tan^{-1} x dx

Solution:

Question 4.

∫x^{2} tan^{-1} x dx

Solution:

Question 5.

∫x^{3} tan^{-1} x dx

Solution:

Let I = ∫x^{3} tan^{-1} x dx

= ∫(tan^{-1} x) . x^{3} dx

Question 6.

∫(log x)^{2} dx

Solution:

Let I = ∫(log x)^{2} dx

Put log x = t

Question 7.

∫sec^{3 }x dx

Solution:

Let I = ∫sec^{3 }x dx

= ∫sec x sec^{2 }x dx

= sec x ∫sec^{2 }x dx – ∫[\(\frac{d}{d x}\)(sec x) ∫sec^{2 }x dx] dx

= sec x tan x – ∫(sec x tan x)(tan x) dx

= sec x tan x – ∫sec x tan^{2 }x dx

= sec x tan x – ∫sec x (sec^{2 }x – 1) dx

= sec x tan x – ∫sec^{3 }x dx + ∫sec x dx

∴ I = sec x tan x – I + log|sec x + tan x|

∴ 2I = sec x tan x + log|sec x + tan x|

∴ I = \(\frac{1}{2}\) [sec x tan x + log|sec x + tan x|] + c.

Question 8.

∫x . sin^{2 }x dx

Solution:

Question 9.

∫x^{3} log x dx

Solution:

Question 10.

∫e^{2x} cos 3x dx

Solution:

Question 11.

∫x sin^{-1} x dx

Solution:

Question 12.

∫x^{2} cos^{-1} x dx

Solution:

Question 13.

\(\int \frac{\log (\log x)}{x} d x\)

Solution:

= t(log t – 1) + c

= (log x) . [log(log x) – 1] + c.

Question 14.

\(\int \frac{t \cdot \sin ^{-1} t}{\sqrt{1-t^{2}}} d t\)

Solution:

Question 15.

∫cos√x dx

Solution:

Let I = ∫cos√x dx

Put √x = t

∴ x = t^{2}

∴ dx = 2t dt

∴ I = ∫(cos t) 2t dt

= ∫2t cos t dt

= 2t ∫cos t dt – ∫[\(\frac{d}{d t}\)(2t) ∫cos t dt]dt

= 2t sin t – ∫2 sin t dt

= 2t sin t + 2 cos t + c

= 2[√x sin√x + cos√x] + c.

Question 16.

∫sin θ . log(cos θ) dθ

Solution:

Let I = ∫sin θ . log (cos θ) dθ

= ∫log(cos θ) . sin θ dθ

Put cos θ = t

∴ -sin θ dθ = dt

∴ sin θ dθ = -dt

= -t log t + t + c

= -cos θ . log(cos θ) + cos θ + c

= -cos θ [log(cos θ) – 1] + c.

Question 17.

∫x cos^{3 }x dx

Solution:

cos 3x = 4 cos^{3 }x – 3 cos x

∴ cos^{3 }x + 3 cos x = 4cos3x

∴ cos^{3 }x = \(\frac{1}{4}\) cos 3x + \(\frac{3}{4}\) cos x

Question 18.

\(\int \frac{\sin (\log x)^{2}}{x} \cdot \log x d x\)

Solution:

Question 19.

\(\int \frac{\log x}{x} d x\)

Solution:

Let I = \(\int \frac{\log x}{x} d x\)

Put log x = t

\(\frac{1}{x}\) dx = dt

∴ I = ∫t dt

= \(\frac{1}{2}\) t^{2} + c

= \(\frac{1}{2}\) (log x)^{2} + c

Question 20.

∫x sin 2x cos 5x dx.

Solution:

Let I = ∫x sin 2x cos 5x dx

sin 2x cos 5x = \(\frac{1}{2}\)[2 sin 2x cos 5x]

= \(\frac{1}{2}\) [sin(2x + 5x) + sin(2x – 5x)]

= \(\frac{1}{2}\) [sin 7x – sin 3x]

∴ ∫sin 2x cos 5x dx = \(\frac{1}{2}\) [∫sin 7x dx – ∫sin 3x dx]

Question 21.

\(\int \cos (\sqrt[3]{x}) d x\)

Solution:

Let I = \(\int \cos (\sqrt[3]{x}) d x\)

II. Integrate the following functions w.r.t. x:

Question 1.

e^{2x} sin 3x

Solution:

Question 2.

e^{-x} cos 2x

Solution:

Question 3.

sin(log x)

Solution:

Question 4.

\(\sqrt{5 x^{2}+3}\)

Solution:

Let I = \(\sqrt{5 x^{2}+3}\) dx

Question 5.

\(x^{2} \sqrt{a^{2}-x^{6}}\)

Solution:

Question 6.

\(\sqrt{(x-3)(7-x)}\)

Solution:

Question 7.

\(\sqrt{4^{x}\left(4^{x}+4\right)}\)

Solution:

Question 8.

(x + 1) \(\sqrt{2 x^{2}+3}\)

Solution:

Let I = ∫(x + 1) \(\sqrt{2 x^{2}+3}\) dx

Let x + 1 = A[\(\frac{d}{d x}\)(2x^{2} + 3)] + B

= A(4x) + B

= 4Ax + B

Comparing the coefficients of x and constant term on both the sides, we get

4A = 1, B = 1

∴ A = \(\frac{1}{4}\), B = 1

Question 9.

\(x \sqrt{5-4 x-x^{2}}\)

Solution:

Let I = ∫\(x \sqrt{5-4 x-x^{2}}\) dx

Let x = A[\(\frac{d}{d x}\)(5 – 4x – x^{2})] + B

= A[-4 – 2x] + B

= -2Ax + (B – 4A)

Comparing the coefficients of x and the constant term on both sides, we get

-2A = 1, B – 4A = 0

Question 10.

\(\sec ^{2} x \sqrt{\tan ^{2} x+\tan x-7}\)

Solution:

Question 11.

\(\sqrt{x^{2}+2 x+5}\)

Solution:

Question 12.

\(\sqrt{2 x^{2}+3 x+4}\)

Solution:

III. Integrate the following functions w.r.t. x:

Question 1.

[2 + cot x – cosec^{2 }x] e^{x}

Solution:

Let I = ∫e^{x} [2 + cot x – cosec^{2 }x] dx

Put f(x) = 2 + cot x

∴ f'(x) = \(\frac{d}{d x}\)(2 + cot x)

= \(\frac{d}{d x}\)(2) + \(\frac{d}{d x}\)(cot x)

= 0 – cosec^{2 }x

= -cosec^{2 }x

∴ I = ∫e^{x} [f(x) + f'(x)] dx

= e^{x} f(x) + c

= e^{x} (2 + cot x) + c.

Question 2.

\(\left(\frac{1+\sin x}{1+\cos x}\right) e^{x}\)

Solution:

Question 3.

\(e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right)\)

Solution:

Let I = ∫\(e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right)\)

Let f(x) = \(\frac{1}{x}\), f'(x) = \(-\frac{1}{x^{2}}\)

∴ I = ∫e^{x} [f(x) + f'(x)] dx

= e^{x} f(x) + c

= e^{x} . \(\frac{1}{x}\) + c

Question 4.

\(\left[\frac{x}{(x+1)^{2}}\right] e^{x}\)

Solution:

Question 5.

\(\frac{e^{x}}{x}\) . [x(log x)^{2} + 2 log x]

Solution:

Question 6.

\(e^{5 x}\left[\frac{5 x \log x+1}{x}\right]\)

Solution:

Let I = ∫\(e^{5 x}\left[\frac{5 x \log x+1}{x}\right]\)

Question 7.

\(e^{\sin ^{-1} x}\left[\frac{x+\sqrt{1-x^{2}}}{\sqrt{1-x^{2}}}\right]\)

Solution:

Question 8.

log(1 + x)^{(1+x)}

Solution :

Let I = ∫log(1 + x)^{(1+x)} dx

Question 9.

cosec (log x)[1 – cot(log x)]

Solution:

Let I = ∫cosec (log x)[1 – cot(log x)] dx

Put log x = t

x = e^{t}

dx = e^{t} dt

I = ∫cosec t (1 – cot t). e^{t} dt

= ∫e^{t} [cosec t – cosec t cot t] dt

= ∫e^{t} [cosec t + \(\frac{d}{d t}\) (cosec t)] dt

= e^{t} cosec t + c ….. [∵ e^{t} [f(t) +f'(t) ] dt = e^{t} f(t) + c ]

= x . cosec(log x) + c.