The value of $\int_0^1 \left[\sin^{-1}\left(\frac{2x}{1+x^2}\right)\right]dx$ is equal to
Step-by-Step Solution
Key Concept: Complex integrals are separated into manageable parts and evaluated using integration by parts and standard inverse trigonometric antiderivatives.
Split the integral: $I = \int_0^1 \sin^{-1}x \, dx + \int_0^1 \frac{e^{x^2+1}}{\cos^2 x} dx$. For the first part, use integration by parts. The second integral becomes $I = [-\tan^{-1}x]_0^1 = -\tan 1$. Combining both parts yields the final answer of $-2\tan 1$.
Correct Answer: -2tan1