How do you integrate int cos2x by integration by parts method?

Choose u=cosx , dv=cosxdv ,

du=−sinxdx

v=sinx . ∫udv

=uv−∫vdu ∫cosx cosxdx

=cosx sinx−∫sinx(−sinx)dx cos 2 xdx

=cosxsinx+∫(1−cos 2 x)dx cos 2 xdx

=cosx sinx+∫1dx−∫cos 2 xdx

take the integral on the right over to the left:

2∫cos 2 xdx=cosx sinx+x

Hence ∫cos 2 xdx = 1/2x +1/2 sinx cosx

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