Question

What is the integral of (1+x)3^x?

Answer 1

`I=(3^x(ln(3)x-1+ln(3)))/(ln(3))^2+C`

Explanation:

We want to solve

`I=int(1+x)3^xdx`

Rewrite the integrand using `a^x=e^(ln(a)x)`

`I=inte^(ln(3)x)dx+intxe^(ln(3)x)dx`

Use integration by parts for the second integral

`intudv=uv-intvdu`

Let `u=x=>du=dx`

And `dv=e^(ln(3)x)dx=>v=1/ln(3)e^(ln(3)x)`

`I=inte^(ln(3)x)dx+1/ln(3)xe^(ln(3)x)-1/ln(3)inte^(ln(3)x)dx`

`color(white)(I)=1/ln(3)xe^(ln(3)x)+(1-1/ln(3))inte^(ln(3)x)dx`

`color(white)(I)=1/ln(3)xe^(ln(3)x)+(1-1/ln(3))1/ln(3)e^(ln(3)x)+C`

`color(white)(I)=(3^x(ln(3)x-1+ln(3)))/(ln(3))^2+C`