Question #6d541

1 Answer
Dec 15, 2017

e^x(x^2+1)+c

Explanation:

Take the integral

inte^x(x+1)^2dx

Expanding the integrand (x+1)^2=x^2+2x+1

int(e^x x^2+2e^x x+e^x)dx

separate

inte^x x^2dx+int2e^x xdx+inte^xdx

For the integrand e^x x^2, integrate by parts,
intfdg=fg-intgdf

f=x^2, dg=e^xdx
df=2xdx g=e^x

e^x x^2-inte^x 2xdx

The integral of e^x=e^x+c

inte^x x^2dx+int2e^x xdx+inte^xdx=e^x x^2-inte^x 2xdx+int2e^x xdx+e^x+c

inte^x x^2dx+int2e^x xdx+inte^xdx=e^x x^2+e^x+c

inte^x x^2dx+int2e^x xdx+inte^xdx=e^x(x^2+1)+c