The value of the integral ∫ e^ 3sin^ - 1 x ( 1 1 - x^ 2 + e^ 3cos^ - 1 x )dx is equal to (where, c & is an arbitrary constant)

Correct answer: C. e3sin^ - 1 x 3 + xe(3π)/(2) + c ∫ e3sin^ - 1 x 1 - x^ 2 dx+∫ e^ 3(sin^ - 1 x + cos^ - 1 x) dx = ∫ e^ 3t dt +∫ e^ (3π)/(2) dx ~(Put sin ^ - 1 x = t) = (e3t)/(3) + e(3π)/(2) ⋅ x + c

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Indefinite Integration

NTA Abhyas 2020 · Mathematics · Indefinite Integration

The value of the integral ∫ e^ 3sin^ - 1 x ( 1 1 - x^ 2 + e^ 3cos^...

The value of the integral

∫ e^{3sin^{- 1}x}\left(\dfrac{1}{\sqrt{1 - x^{2}}} + e^{3cos^{- 1}x}\right)dx

is equal to
(where,

c

& is an arbitrary constant)

Options

A $\dfrac{e^{3\sqrt{sin^{- 1}x}}}{3} + xe^{\dfrac{3\pi}{2}} + c$
B $e^{\sqrt{sin^{- 1}x}} + e^{\pi / 2} + c$
C $\dfrac{e3sin^{- 1}x}{3} + xe\dfrac{3\pi}{2} + c$
D $e^{\dfrac{\pi}{2}} + e^{x\left(\dfrac{\pi}{2}\right)} + c$

Correct answer

C. $\dfrac{e3sin^{- 1}x}{3} + xe\dfrac{3\pi}{2} + c$

Step-by-step solution

∫ \dfrac{e3sin^{- 1}x}{1 - x^{2}}dx+∫ e^{3\left(sin^{- 1}x + cos^{- 1}x\right)}dx

= ∫ e^{3t}dt +∫ e^{\dfrac{3\pi}{2}}dx

~\left(Put{sin}^{- 1} x = t\right)

<br />

= \dfrac{e3t}{3} + e\dfrac{3\pi}{2} ⋅ x + c

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