How do you integrate #int e^x sin x dx # using integration by parts? Calculus Techniques of Integration Integration by Parts 1 Answer Lovecraft Jan 7, 2016 #inte^xsin(x)dx = (e^xsin(x) - e^xcos(x))/2 + c# Explanation: Say #dv = e^x# so #v = e^x#, #u = sin(x)# so #du = cos(x)# #inte^xsin(x)dx = e^xsin(x) - inte^(x)cos(x)dx# Say #dv = e^x# so #v = e^x#, #u = cos(x)# so #du = -sin(x)# #inte^xsin(x)dx = e^xsin(x) - (e^xcos(x) +inte^xsin(x)dx)# #inte^xsin(x)dx = e^xsin(x) - e^xcos(x) -inte^xsin(x)dx# #2inte^xsin(x)dx = e^xsin(x) - e^xcos(x)# #inte^xsin(x)dx = (e^xsin(x) - e^xcos(x))/2 + c# Answer link Related questions How do I find the integral #int(x*ln(x))dx# ? How do I find the integral #int(cos(x)/e^x)dx# ? How do I find the integral #int(x*cos(5x))dx# ? How do I find the integral #int(x*e^-x)dx# ? How do I find the integral #int(x^2*sin(pix))dx# ? How do I find the integral #intln(2x+1)dx# ? How do I find the integral #intsin^-1(x)dx# ? How do I find the integral #intarctan(4x)dx# ? How do I find the integral #intx^5*ln(x)dx# ? How do I find the integral #intx*2^xdx# ? See all questions in Integration by Parts Impact of this question 1512 views around the world You can reuse this answer Creative Commons License