Question

Assertion :$\int e^{{\tan }^{-1}x}\left(\frac{1+x+x^2}{1+x^2}\right)dx=x{\tan }^{-1}x+C$ Reason: $\int e^t(f\left(t\right)+f^{\prime }\left(t\right))dt=e^{t}f\left(t\right)+C$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Reason is true (theory)
$I=\int e^{{\tan }^{-1}}x\left(\frac{1+x+x^2}{1+x^2}\right)dx{\tan }^{-1}x=t\Rightarrow x=\tan t\Rightarrow 1+x^2={\sec }^{2}tI=\int e^t(\tan t+{\sec }^{2}t)dt=e^t\tan t+C=x{e}^{{\tan }^{-1}x}+C$