Question

Найти интеграл,используя основные методы интегрирования:
$\int\limits {x^{2} *e^{5x+2} } \, dx$

Answer 1

$\displaystyle \int x^2 e^{5x+2}dx=\left\{\begin{array}{ccc}u=x^2~~~ du=2xdx\\ \\ e^{5x+2}dx=dv~~~ v=\dfrac{1}{5}e^{5x+2}\end{array}\right\}=\dfrac{x^2}{5}e^{5x+2}-\\ \\ -\int \dfrac{1}{5}e^{5x+2}\cdot 2xdx=\dfrac{x^2}{5}e^{5x+2}-\dfrac{2}{5}\int xe^{5x+2}dx=\\ \\ \\ =\left\{\begin{array}{ccc}u=x~~~~ du=dx\\ \\ dv=e^{5x+2}dx~~~~ v=\dfrac{1}{5}e^{5x+2}\end{array}\right\}=\dfrac{x^2}{5}e^{5x+2}-\dfrac{2}{5}\Bigg(\dfrac{x}{5}e^{5x+2}-\\ \\ \\ -\int \dfrac{1}{5}e^{5x+2}dx\Bigg)=\dfrac{x^2}{5}e^{5x+2}-\dfrac{2x}{25}e^{5x+2}+\dfrac{2}{25}\cdot \dfrac{1}{5}e^{5x+2}+C=$

$\displaystyle =e^{5x+2}\Big(\dfrac{x^2}{2}-\dfrac{2x}{25}+\dfrac{2}{125}\Big)+C=\dfrac{e^{5x+2}}{125}\Big(25x^2-10x+2\Big)+C$