Question

$\int e^x\left[f\right(x)+f^{\prime }(x\left)\right]dx$ is equal to

• A. $e^{x}f\left(x\right)+c$
• B. $e^x+c$
• C. $e^x{f}^{\prime }\left(x\right)+c$
• D. None of these

Answer 1

The correct option is A $e^{x}f\left(x\right)+c$

Let $I=\int e^x\left[f\right(x)+f^{\prime }(x\left)\right]dx$
$=\int e^{x}f\left(x\right)dx+\int e^x{f}^{\prime }\left(x\right)dx$
$=e^{x}f\left(x\right)-\int e^x{f}^{\prime }\left(x\right)dx+\int e^x{f}^{\prime }\left(x\right)dx$
$=e^{x}f\left(x\right)+c$