Question

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

Answer 1

`f(x)=(xe^(2x))/2 -e^(2x)/4 +(3x^4)/4 +3/(4e^2) -3/4`

Explanation:

`int xe^(2x) +3x^3 dx= int xe^(2x) dx +int 3x^3 dx` +C

=`(xe^(2x)) /2 -int e^(2x) /2dx + (3x^4) /4`+C

f(x)=`(xe^(2x))/2 -e^(2x)/4 +(3x^4)/4 +C`

Now letting x= -1, `f(-1)=1= -1/(2e^2) -1/(4e^2) +3/4 +C`

This gives C= `3/(4e^2)-3/4`

Therefore `f(x)=(xe^(2x))/2 -e^(2x)/4 +(3x^4)/4 +3/(4e^2) -3/4`