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 }
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 \begin{align} \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{align} }
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{Now evaluate}\int\sin^4(x)dx }
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 \begin{align} \int\sin^4(x)dx &= - \frac{1}{4}\cos(x)\sin^3(x) + \frac{3}{4} \int\sin^2(x)dx \\[2ex] &= - \frac{1}{4}\cos(x)\sin^3(x) + \frac{3}{4}(-\frac{1}{4}\sin(2x) + \frac{x}{2}) \\[2ex] &= - \frac{1}{4}\cos(x)\sin^3(x) - \frac{3}{16}\sin(2x) + \frac{3}{8}x + c \\[2ex] \end{align} }