5.4 Indefinite Integrals and the Net Change Theorem/3

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int \cos ^{3}xdx=\sin {x}-{\frac {1}{3}}\sin ^{3}x+C\\[2ex]{\frac {d}{dx}}{[\sin {x}-{\frac {1}{3}}\sin ^{3}{x}+C]}\\[2ex]&={\cos {x}-{\frac {1}{3}}\cdot 3\sin ^{2}{x}\cos {x}+0}\\[2ex]&=\cos {x}-\sin ^{2}{x}\cos {x}\\[2ex]&=\cos {x}-(1-cos^{2}(x))\cos {x}\\[2ex]&=\cos ^{3}{x}\end{aligned}}}

Note: $1-\cos ^{2}(x)=\sin ^{2}(x)$