What is f(x) = int xe^(2x) + x^3 dx if f(-3 ) = 1 ?

1 Answer
Mar 31, 2018

f(x)=1/4{(2x-1)e^(2x)+x^4+7(e^-6-11)}.

Explanation:

f(x)=int(xe^(2x)+x^3)dx,

=intxe^(2x)dx+intx^3dx,

=intxe^(2x)dx+x^4/4,

=[x*inte^(2x)dx-int(d/dx(x)inte^(2x)dx)dx]+x^4/4...[because, IBP],

=x*e^(2x)/2-int(1*e^(2x)/2)dx+x^4/4,

rArr f(x)=(xe^(2x))/2-1/2*e^(2x)/2+x^4/4+C............(ast).

To determine C, we use the given condition : f(-3)=1.

f(-3)=1, &, (ast) rArr 1=(-3*e^-6)/2-e^-6/4+81/4+C.

:. C=7/4(e^-6-11).

rArr f(x)=1/4{(2x-1)e^(2x)+x^4+7(e^-6-11)}.

Enjoy Maths.!