Question

Evaluate: ∫4 𝑥𝑒^𝑥 dx.

Answer 1

$∫4 𝑥𝑒^𝑥 dx$

Apply linearity

$4\smallint{xe^x}dx$

Now solving $\smallint{xe^x}dx$

Integrate by parts $\smallint fg'=fg- \smallint f'g$

$f=x$ $g'=e^x$

$f'=1$ $g=e^x$

$=xe^x-\smallint e^xdx$

Now solving $\smallint e^xdx$

Apply exponential rule,

$\smallint ae^x=\frac{a^x}{ln(a)}$ With $a=e$

$=e^x$

Plug in solved integrals

$=xe^x-\smallint e^xdx =xe^x- e^x$

Plug in solved integrals

$4\smallint xe^xdx=4 xe^x-4e^x$

The solved problem

$∫4 𝑥𝑒^𝑥 dx$ $=4xe^x-4e^x+c$

$=4(x-1)e^x+c$