Question

How do you integrate `int x*5^x` by integration by parts method?

Answer 1

`intx*5^x` `dx=`

Explanation:

`intx*5^x` `dx`

Let `f=x`, then `df=dx`

And `dg=5^x` `dx`, then `g=5^x /ln5`

`intx*5^x` `dx=(x*5^x)/ln5-1/ln5int5^x` `dx`

`=(x*5^x)/ln5-5^x/ln^2 5+"c"`

Taking out a factor of `1/ln^2 5` yields:

`intx*5^x` `dx=1/ln^2 5[(x*5^x)ln5-5^x]+"c"`

How to integrate `5^x` `dx`

`int5^x` `dx`

Let `u=5^x`

Then `du=5^xln5` `dx`

And `dx=1/(5^xln5)` `du`

`thereforeint5^x` `dx=int5^x xx 1/(5^xln5)` `du=1/ln5int1` `du=u/ln5+"c"`
`=5^x/ln5+"c"`