How do you find the antiderivative of #(cosx)(e^x)#?

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Answer 1 · 2017-06-08T00:10:18

#int e^x cosx dx = (e^xcosx + e^xsinx)/2 +C#

Start by integrating by parts:

#int e^x cosx dx = int cosx d(e^x) = e^xcosx + int e^xsinx dx#

integrate again by parts:

#int e^x cosx dx = e^xcosx + e^xsinx - int e^xcosx dx#

The same integral appears now on both sides of the equation so we can solve for it:

#2int e^x cosx dx = e^xcosx + e^xsinx +C#

and finally:

#int e^x cosx dx = (e^xcosx + e^xsinx)/2 +C#