I am encountering a paradox when calculating the integral ##\int sin(x)\cos(x)\,dx## with(adsbygoogle = window.adsbygoogle || []).push({}); integration by parts:

Defining ##u = sin(x), v' = cos(x)##:

##\int sin(x)cos(x) dx = sin^2(x) - \int cos(x) sin(x) dx##

##\Leftrightarrow \int sin(x) * cos(x) dx = +1/2*sin^2(x)##.

On the other hand, defining ##u = cos(x), v' = sin(x)##:

##\int sin(x)cos(x) dx = -cos^2(x) - \int cos(x) sin(x) dx##

##\Leftrightarrow \int sin(x) * cos(x) dx = -1/2*cos^2(x)##.

##1/2*sin^2(x) \neq -1/2*cos^2(x)##, and I know that the second one is the correct one. But I can't find an error in my upper calculation. Does somebody see the error I made?

Thanks!

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# Integrating sin(x)*cos(x) paradox

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