Question

How can I evaluate the integral `intx^2e^(4x)dx`?

Answer 1

I would use integration by parts where you have:

`intf(x)*g(x)dx=F(x)*g(x)-intF(x)*g'(x)dx`

Where:

`F(x)=intf(x)dx`

`g'(x)` is the derivative of `g(x)`

I'll choose:

`f(x)=e^(4x)` and
`g(x)=x^2`

Integrating you get:

`intx^2e^(4x)dx=x^2e^(4x)/4-int2xe^(4x)/4dx=`
`=x^2e^(4x)/4-intxe^(4x)/2dx=`
`=x^2e^(4x)/4-1/2intxe^(4x)dx=` and by parts again:
`=x^2e^(4x)/4-1/2[xe^(4x)/4-int1e^(4x)/4dx]=`
`=x^2e^(4x)/4-xe^(4x)/8+e^(4x)/32+c=`
`=e^(4x)(x^2/4-x/8+1/32)+c`