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

1 Answer
Mar 11, 2018

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