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
I=inte^(ln(3)x)dx+intxe^(ln(3)x)dx
Use integration by parts for the second integral
intudv=uv-intvdu
Let
And
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