Question

x*e^sqrtx dx?

Answer 1

\(\large\color{#000000 }{ \displaystyle \int xe^\sqrt{x}dx }\)

\(\large\color{#a11111 }{ \displaystyle u=\sqrt{x} }\)
therefore, \(\large\color{#a11111 }{ \displaystyle x=u^2 }\)

\(\large\color{#a11111 }{ \displaystyle du=\frac{1}{2\sqrt{x}}dx\quad \Longrightarrow \quad 2\sqrt{x}~du=dx\quad \Longrightarrow \quad 2u~du=dx}\)

\(\large\color{#000000 }{ \displaystyle \int 2u^3e^udu }\)

Answer 2

From here you can do integration by parts 3 times to get rid of \(u^3\) in front of \(e^u\), and then don't forget to substitute back \(u=\sqrt{x}\) accordingly.

Answer 3

Alternatively, you can use the following series
(if you have learned series techniques of integration)

\(\large\color{#000000}{ \displaystyle e^u=\sum_{ n=0 }^{ \infty } ~ \frac{ u^n }{n!}}\)

Answer 4

Ohh, thank u