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

1 Answer
Dec 11, 2016

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