Question

Let $\int e^x\left\{f\right(x)-f^{\prime }(x\left)\right\}dx=\varphi \left(x\right).$ Then $\int e^{x}f\left(x\right)dx$ is

• A. $\varphi \left(x\right)=e^{x}f\left(x\right)$
• B. $\varphi \left(x\right)-e^{x}f\left(x\right)$
• C. $\frac{1}{2}\left\{\varphi \right(x)+e^{x}f(x\left)\right\}$
• D. $\frac{1}{2}\left\{\varphi \right(x)+e^x{f}^{\prime }(x\left)\right\}$

Answer 1

The correct option is C $\frac{1}{2}\left\{\varphi \right(x)+e^{x}f(x\left)\right\}$

Given $\int e^x\left\{f\right(x)-f^{\prime }(x\left)\right\}dx=\varphi \left(x\right).$ ...(1)
Consdier, $\int e^x\left\{f\right(x)-f^{\prime }(x\left)\right\}dx$
$=\int e^{x}f\left(x\right)dx-\int e^x{f}^{\prime }\left(x\right)dx$
$=e^{x}f\left(x\right)-2\int e^x{f}^{\prime }\left(x\right)dx$
So, by (1),

$e^{x}f\left(x\right)-2\int e^x{f}^{\prime }\left(x\right)dx=\varphi \left(x\right)$

$\int e^{x}f\left(x\right)dx=\frac{1}{2}\left\{\varphi \right(x)+e^{x}f(x\left)\right\}$