5.5 The Substitution Rule/35

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 \frac{sin 2 x}{1 + cos^2x} dx \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 \begin{align} let \; u &=1 + cos^2x \\[2ex] du &= 2cosx \;\cdot (-sinx)\;dx \\[2ex] -du & = 2sin(x)cos(x)\;dx & \\[2ex] -du & = sin(2x)dx & \end{align} }

${\begin{aligned}\ \int {\frac {sin2x}{1+cos^{2}x}}dx\;=\;-\ \int {\frac {du}{u}}\;=\;-ln(1+u)+c\;=\;-|1+cos^{2}x|+c\end{aligned}}$