Just in case a picture helps with the anti-derivative...

... where (key in spoiler) ...

Spoiler:

After the sub, cos as the inner derivative is cancelled by another cos produced courtesy of Pythag.

Edit: the definite integral...

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!