Evaluate the integral using integration by parts.
-$\int \sin ^ { - 1 } x$ dx
A) $\sin ^ { - 1 }$ x + $\sqrt { 1 - x ^ { 2 } }$ + C
B) x + $\sqrt { 1 - x ^ { 2 } }$ + C
C) x $\sin ^ { - 1 }$ x + $\sqrt { 1 - x ^ { 2 } }$ + C
D) $\sin ^ { - 1 }$ x - $\sqrt { 1 - x ^ { 2 } }$ + C

Question

Quizplus · Accepted Answer

Integration by parts formula is $\int u dv = uv - \int v du$. Let $u = \sin^{-1}x$ and $dv = dx$, then $du = \frac{1}{\sqrt{1-x^2}}dx$ and $v = x$. Applying the formula gives $x\sin^{-1}x - \int x\left(\frac{1}{\sqrt{1-x^2}}\right)dx$. The integral of the second term results in $\sqrt{1-x^2}$, leading to the answer $x\sin^{-1}x + \sqrt{1-x^2} + C$.

Evaluate the Integral Using Integration by Parts ∫sin⁡−1x\int \sin ^ { - 1 } x∫sin−1x

Evaluate the Integral Using Integration by Parts $\int \sin ^ { - 1 } x$