7.1 Integration By Parts/43

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{Prove with the reduction formula that} \int\sin^2(x)dx = \frac{x}{2} - \frac{sin(2x)}{4} + C }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \text{reduction formula:} \int\sin^ndx= -\frac{1}{n}\cos(x)\sin^{n-1}(x) + \frac{n-1}{n}\int\sin^{n-2}(x)dx } ${\begin{aligned}\int \sin ^{2}(x)dx&=-{\frac {1}{2}}\cos(x)\cdot \sin ^{2-1}x+{\frac {2-1}{2}}\int \sin ^{0}(x)dx\\[2ex]&=-{\frac {1}{2}}\cos(x)\sin(x)+{\frac {1}{2}}x+c\\[2ex]&=-{\frac {1}{2}}\cdot 2\cdot {\frac {1}{2}}\sin(x)\cos(x)+{\frac {x}{2}}+c\\[2ex]&=-{\frac {1}{4}}\sin(2x)+{\frac {x}{2}}+c\\[2ex]\end{aligned}}$