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\begin{aligned} \int \sin ^{3}(5 x) \mathrm{dx} &=\frac{-\sin ^{2}(5 x) \cdot \cos (5 x)}{3 \cdot 5}+\frac{2}{3} \int \sin (5 x) \mathrm{dx} \\ &=\frac{-\sin ^{2}(5 x) \cos (5 x)}{15}-\frac{2}{3} \frac{1}{5} \cos (5 x)+\mathrm{C} \end{aligned}