How do you integrate $int sinx*e^-x$ by integration by parts method?

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Answer 1 · 2018-05-25T13:43:23

$int sinx e^(-x) dx = -(e^(-x)(sinx +cosx ))/2+ C$

Integrate by parts:

$int sinx e^(-x) dx = int sinx d/dx(-e^(-x)) dx$

$int sinx e^(-x) dx = -e^(-x)sinx + int e^(-x) d/dx(sinx ) dx$

$int sinx e^(-x) dx = -e^(-x)sinx + int e^(-x) cosx dx$

and then again:

$int sinx e^(-x) dx = -e^(-x)sinx + int cosx d/dx (-e^(-x))dx$

$int sinx e^(-x) dx = -e^(-x)sinx -e^(-x)cosx + int e^(-x) d/dx(cosx )dx$

$int sinx e^(-x) dx = -e^(-x)sinx -e^(-x)cosx - int sinx e^(-x) dx$

The same integral appears on both sides and we can solve fro it:

$2int sinx e^(-x) dx = -e^(-x)sinx -e^(-x)cosx + C$

$int sinx e^(-x) dx = -(e^(-x)(sinx +cosx ))/2+ C$

How do you integrate int sinx*e^-xint sinx*e^-x by integration by parts method?