Expert Answer to "Evaluate the integral. ∫sin−1x1−x2"

ringearV

Answered question

2020-12-07

Evaluate the integral.

\int \frac{\sin^{- 1} x}{\sqrt{1 - x^{2}}}

Faiza Fuller

Skilled2020-12-08Added 108 answers

Step 1
Consider the provided integral,

\int \frac{\sin^{- 1} x}{\sqrt{1 - x^{2}}} d x

Evaluate the integrals.
Apply the Substitution method,

u = \sin^{- 1} x \Rightarrow d u = \frac{1}{\sqrt{1 - x^{2}}} d x

Step 2
Therefore,

\int \frac{\sin^{- 1} x}{\sqrt{1 - x^{2}}} d x = \int u d u

= \frac{u^{1 + 1}}{1 + 1} + C

= \frac{u^{2}}{2} + C

Substitute back

u = \sin^{- 1} x

\int \frac{\sin^{- 1} x}{\sqrt{1 - x^{2}}} d x = \frac{{(\sin^{- 1} x)}^{2}}{2} + C

= \frac{1}{2} {(\sin^{- 1} x)}^{2} + C

Hence.

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