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1.
$\int\limits_{0}^{\frac{\pi}{2}}\frac{sin^5x}{cosx+1}dx=16\int\limits_{0}^{\frac{\pi}{2}}sin^5\frac{x}{2}cos^3\frac{x}{2}dx=32\int\limits_{0}^{\frac{\sqrt{2}}{2}}sin^5\frac{x}{2}(1-sin^2\frac{x}{2})d(sin\frac{x}{2})$
$=32\int\limits_{0}^{\frac{\sqrt{2}}{2}}(t^5-t^7)dt=\frac{5}{12}$
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