Question

Let $\int e^x\{f\left(x\right)-f^{\prime }\left(x\right)\}dx=\varphi \left(x\right).$ 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. $\frac{1}{2}\{\varphi \left(x\right)+e^{x}f\left(x\right)\}$
• D. $\frac{1}{2}\{\varphi \left(x\right)+e^x{f}^{\prime }\left(x\right)\}$

Answer 1

Answer: (C)

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

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