What is f(x) = int xe^(2x) + 3x^3 dxf(x)=∫xe2x+3x3dx if f(-1 ) = 1 f(−1)=1?

1 Answer
bp
Dec 11, 2016

f(x)=(xe^(2x))/2 -e^(2x)/4 +(3x^4)/4 +3/(4e^2) -3/4f(x)=xe2x2−e2x4+3x44+34e2−34

Explanation:

int xe^(2x) +3x^3 dx= int xe^(2x) dx +int 3x^3 dx∫xe2x+3x3dx=∫xe2xdx+∫3x3dx +C

=(xe^(2x)) /2 -int e^(2x) /2dx + (3x^4) /4xe2x2−∫e2x2dx+3x44+C

f(x)=(xe^(2x))/2 -e^(2x)/4 +(3x^4)/4 +Cxe2x2−e2x4+3x44+C

Now letting x= -1, f(-1)=1= -1/(2e^2) -1/(4e^2) +3/4 +Cf(−1)=1=−12e2−14e2+34+C

This gives C= 3/(4e^2)-3/434e2−34

Therefore f(x)=(xe^(2x))/2 -e^(2x)/4 +(3x^4)/4 +3/(4e^2) -3/4f(x)=xe2x2−e2x4+3x44+34e2−34

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