Question

Is there any formula like integral $\int e^x\left(f\left(x\right)f^{\prime }\left(x\right)\right)dx=e^{x}f\left(x\right)+C.$

Answer 1

$\begin{array}{c}\int e^x\left(f\left(x\right)+f^{\prime }\left(x\right)d\right)dx=\int e^{x}f\left(x\right)+e^x{f}^{\prime }\left(x\right)dx\\ Letg\left(x\right)=e^x\\ g^{\prime }\left(x\right)=e^x\\ \int e^x\left(f\left(x\right)+f^{\prime }\left(x\right)\right)dx=\int g^{\prime }\left(x\right)f\left(x\right)+f^{\prime }\left(x\right)g\left(x\right)dx\\ usetheproductruleandfind\\ \int g^{\prime }\left(x\right)f\left(x\right)+f^{\prime }\left(x\right)g\left(x\right)dx=g\left(x\right)f\left(x\right)+C\\ \int e^x\left(f\left(x\right)f^{\prime }\left(x\right)\right)dx=e^{x}f\left(x\right)+C.\end{array}$