Question

How do you integrate `f(x)=3^xe^(-x+1)` using the product rule?

Answer 1

`inte^(xln3-x+1)dx=e^(xln3-x+1)/(ln3-1)`

Explanation:

We can rewrite it like this`f(x)=e^(xln3)e^(-x+1)` so that we can write `inte^(xln3)e^(-x+1)dx` and integrating by parts we have
`inte^(xln3)e^(-x+1)dx=`
`=-e^(-x+1)e^(xln3)-int-e^(-x+1)ln3e^(xln3)dx=`
`-e^(-x+1)e^(xln3)+ln3inte^(xln3-x+1)dx`
`inte^(xln3-x+1)dx=e^(xln3-x+1)/(ln3-1)`