Evaluate the integral.
-Use the formula $\int \mathrm { f } ^ { - 1 } ( \mathrm { x } ) \mathrm { dx } = \mathrm { xf } ^ { - 1 } ( \mathrm { x } ) - \int x \left( \frac { \mathrm { d } } { \mathrm { dx } } \mathrm { f } ^ { - 1 } ( \mathrm { x } ) \right) \mathrm { dx }$ to evaluate the integral. $\int \sin ^ { - 1 } x d x$
A) $x \sin ^ { - 1 } x + \sqrt { 1 - x ^ { 2 } } + C$
B) $x \sin ^ { - 1 } x + x + C$
C) $x \sin ^ { - 1 } x + \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } + C$
D) $x \sin ^ { - 1 } x - \sqrt { 1 - x ^ { 2 } } + C$

Question

Quizplus · Accepted Answer

Using the given formula, we substitute $f^{-1}(x) = \sin^{-1}(x)$. The derivative of $\sin^{-1}(x)$ is $\frac{1}{\sqrt{1-x^2}}$, so the integral becomes $x\sin^{-1}(x) - \int \frac{x}{\sqrt{1-x^2}} dx$. Solving the integral gives $x\sin^{-1}(x) + \sqrt{1-x^2} + C$.

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