How can I integrate e^x sin(x)?

This integral is a particularly difficult integral that has shown up in STEP questions before. It's not hard because of how difficult the actual calculations are, but more because of how hard it is to spot the 'trick' to doing this integral.
The trick is integration by parts twice. You can integrate by parts by letting u=e^x and v'=sin(x). You end up with the integral of e^x cos(x), which you integrate with exactly the same method. Then, you end up with, if I is the integral of e^x sin(x),
I=e^x sin(x) - e^x cos(x) - I + c, which implies that I=e^x (sin(x)-cos(x)) + c.

Answered by Lawrence H. STEP tutor

1888 Views

See similar STEP University tutors

Related STEP University answers

All answers ▸

Show that i^i = e^(-pi/2).


What do integrals and derivatives actually do/mean?


Suppose that 3=2/x(1)=x(1)+(2/x(2))=x(2)+(2/x(3))=x(3)+(2/x(4))+...Guess an expression, in terms of n, for x(n). Then, by induction or otherwise, prove the correctness of your guess.


Find 100 consecutive natural numbers, each of which is composite


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2024

Terms & Conditions|Privacy Policy