Question

How to integrate `intx^2e^(-5x)dx` by using integration by parts?

`intx^2e^(-5x)dx`

Answer 1

`I=-e^(-5x)[x^2/5+(2x)/25+2/125]+c`

Explanation:

Here,

`I=intx^2e^(-5x)dx`

`"Using "color(blue) "Integration by Parts":`

`int(u*v)dx=uintvdx-int(u'intvdx)dx`

Let, `u=x^2 and v=e^(-5x)`

`=>u'=2x and intvdx=(e^(-5x))/(_5)`

`I=x^2(e^(-5x)/(-5))-int2x(e^(-5x)/(-5))dx`

`=-x^2/5e^(-5x)+2/5intxe^(-5x)dx`

Again,`"Using "color(blue) "Integration by Parts":`in second
integral

`=-x^2/5e^(-5x)+2/5[x(e^(_5x)/(-5))-int(e^(-5x)/(-5))dx]`

`=-x^2/5e^(-5x)+2/5[-x/5e^(-5x)+1/5(e^(-5x)/(-5))]+c`

`=-x^2/5e^(-5x)-(2x)/25e^(-5x)-2/125e^(-5x)+c`

`I=-e^(-5x)[x^2/5+(2x)/25+2/125]+c`