Integrate $sin (a x + b) cos (a x + b)$

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Solution

$I = \int sin (a x + b) cos (a x + b) d x$
$\begin{matrix} sin (a x + b) = t a cos (a x + b) d x = d t cos (a x + b) d x = \frac{1}{a} d t I = \frac{1}{a} \int t d t = \frac{1}{a} \frac{t^{2}}{2} + c = \frac{1}{2 a} {sin}^{2} (a x + b) + c \end{matrix}$

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