Question

The value of the integral $\int e^x\left\{f\right(x)+f^{\prime }(x\left)\right\}dx$ is

• A. $e^x{f}^{\prime }\left(x\right)+c$
• B. $e^{x}f\left(x\right)+c$
• C. $\frac{e^x}{f\left(x\right)}+c$
• D. $\frac{e^x}{f^{\prime }\left(x\right)}+c$

Answer 1

Answer: (B)

$e^{x}f\left(x\right)+c$

$\int e^{x}f\left(x\right)dx$ + $\int e^x{f}^{\prime }\left(x\right)dx$

Now taking $e^x$ as first function and $f^{\prime }\left(x\right)$ as the second function in integration by parts, we have,

$\Rightarrow$

$\int e^{x}f\left(x\right)dx$ + $e^{x}f\left(x\right)$ - $\int e^{x}f\left(x\right)dx$

$=e^{x}f\left(x\right)+c$