Write a value
Question:

Write a value of $\int \mathrm{e}^{\mathrm{ax}} \mid$ a $f(x)+f^{\prime}(x) \mid d x$.

Solution:

given $\int e^{a x}\left|a f(x)+f^{\prime}(x)\right| d x$

$=a \int e^{a x} f(x) d x+\int e^{a x} f^{\prime}(x) d x$

$=a\left[f(x) \frac{e^{a x}}{a}-\int f^{\prime}(x) \frac{e^{a x}}{a} d x\right]+\int e^{a x} f^{\prime}(x) d x$

$=f(x) e^{a x}+c$