What is f(x) = int xe^(x+2)-x dx if f(-1) = 2 ? Calculus Techniques of Integration Evaluating the Constant of Integration 1 Answer Ratnaker Mehta Aug 22, 2016 f(x)=xe^(x+2)-e^(x+2)-x^2/2+5/2+2e. Explanation: f(x)=int(xe^(x+2)-x)dx=intxe^(x+2)dx-intxdx=I-x^2/2, where, I=intxe^(x+2)dx. To evaluate I, we use the Rule of Integration by Parts : intuvdx=uintvdx-int[(du)/dxintvdx]dx. We take, u=x rArr (du)/dx=1, and, v=e^(x+2) rArr intvdx=e^(x+2). Hence, I=xe^(x+2)-inte^(x+2)dx=xe^(x+2)-e^(x+2) :. f(x)=xe^(x+2)-e^(x+2)-x^2/2+C...........(1). To determine C, we use the cond. : f(-1)=2 in (1). :. -e-e-1/2+C=2 rArr C=5/2+2e. Therefore, (1) gives, f(x)=xe^(x+2)-e^(x+2)-x^2/2+5/2+2e. Enjoy Maths.! Answer link Related questions How do you find the constant of integration for intf'(x)dx if f(2)=1? What is a line integral? What is f(x) = int x^3-x if f(2)=4 ? What is f(x) = int x^2+x-3 if f(2)=3 ? What is f(x) = int xe^x if f(2)=3 ? What is f(x) = int x - 3 if f(2)=3 ? What is f(x) = int x^2 - 3x if f(2)=1 ? What is f(x) = int 1/x if f(2)=1 ? What is f(x) = int 1/(x+3) if f(2)=1 ? What is f(x) = int 1/(x^2+3) if f(2)=1 ? See all questions in Evaluating the Constant of Integration Impact of this question 1480 views around the world You can reuse this answer Creative Commons License