Evaluate the integral.
\[
\int \frac{e^{\sin ^{-1} x} d x}{\sqrt{1-x^{2}}}
\]
Answer
Final Answer: The integral of \(\frac{e^{\sin ^{-1} x}}{\sqrt{1-x^{2}}}\) with respect to \(x\) is \(\boxed{e^{\sin^{-1}x} + C}\), where \(C\) is the constant of integration.
Step 1 ：Let's start by substituting \(u = \sin^{-1}x\). This implies that \(du = \frac{dx}{\sqrt{1-x^{2}}}\).
Step 2 ：The integral then becomes \(\int e^u du\), which is a simple integral to solve.
Step 3 ：The integral of \(e^u\) with respect to \(u\) is \(e^u\).
Step 4 ：Now we substitute \(u\) back with \(\sin^{-1}x\) to get the final answer.
Step 5 ：Final Answer: The integral of \(\frac{e^{\sin ^{-1} x}}{\sqrt{1-x^{2}}}\) with respect to \(x\) is \(\boxed{e^{\sin^{-1}x} + C}\), where \(C\) is the constant of integration.
