Question

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

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Consider, $\int \frac{x{e}^x}{{(1+x)}^2}dx$
$=\int \frac{(x+1)-1}{{(1+x)}^2}dx$
$=\int e^x\{\frac{1}{(1+x)}-\frac{1}{{(1+x)}^2}\}dx$
$=\int e^x(f\left(x\right)+f^{\prime }\left(x\right))dx$
where $f\left(x\right)=\frac{1}{(1+x)}$
$f^{\prime }\left(x\right)=-\frac{1}{(1+x{)}^2}$

So, $I=\frac{e^x}{x+1}+C$