How can I evaluate the integral #intx^2e^(4x)dx#?

1 Answer
Jan 31, 2015

I would use integration by parts where you have:

#intf(x)*g(x)dx=F(x)*g(x)-intF(x)*g'(x)dx#

Where:

#F(x)=intf(x)dx#

#g'(x)# is the derivative of #g(x)#

I'll choose:

#f(x)=e^(4x)# and
#g(x)=x^2#

Integrating you get:

#intx^2e^(4x)dx=x^2e^(4x)/4-int2xe^(4x)/4dx=#
#=x^2e^(4x)/4-intxe^(4x)/2dx=#
#=x^2e^(4x)/4-1/2intxe^(4x)dx=# and by parts again:
#=x^2e^(4x)/4-1/2[xe^(4x)/4-int1e^(4x)/4dx]=#
#=x^2e^(4x)/4-xe^(4x)/8+e^(4x)/32+c=#
#=e^(4x)(x^2/4-x/8+1/32)+c#