∫ 1 8 + 2 x - x 2 d x =

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Question: Answered & Verified by Expert

\int \frac{1}{\sqrt{8 + 2 x - x^{2}}} d x =

MathematicsIndefinite IntegrationMHT CETMHT CET 2016

Options:

A $\frac{1}{3} \sin^{- 1} (\frac{x - 1}{3}) + c$
B $\sin^{- 1} (\frac{x + 1}{3}) + c$
C $\frac{1}{3} \sin^{- 1} (\frac{x + 1}{3}) + c$
D $\sin^{- 1} (\frac{x - 1}{3}) + c$

Solution:

1715 Upvotes Verified Answer

The correct answer is:

\sin^{- 1} (\frac{x - 1}{3}) + c

Let

I = \int \frac{d x}{\sqrt{8 + 2 x - x^{2}}}

\Rightarrow I = \int \frac{d x}{\sqrt{9 + 2 x - x^{2} - 1}}

\Rightarrow I = \int \frac{d x}{3^{2} - {(x - 1)}^{2}}

∴ I = \sin^{- 1} (\frac{x - 1}{3}) + c

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