$\int {(\sqrt{sin x} + cos x)}^{2} d x$

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Solution

$\int (\sqrt{sin x} + cos x)^{2} d x = \int sin x d x + \int {cos}^{2} x d x - 2 \int \sqrt{sin x} cos x d x$
$= - cos x + \frac{x}{2} + \frac{sin 2 x}{2} - \frac{4}{3} (sin x)^{4 / 3} + c$

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