5.5 The Substitution Rule/35: Difference between revisions
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<math>\begin{align} \ \int \frac{sin 2 x}{1 + cos^2x} dx | <math>\begin{align} \ \int \frac{sin 2 x}{1 + cos^2x} dx | ||
\end{align} | |||
</math> | |||
<math>\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} | |||
</math> | |||
<math>\begin{align} \ \int \frac{sin 2 x}{1 + cos^2x} dx\;=\; -\ \int \frac {du}{u}\;=\;-ln( 1 + u) + c\;=\;-| 1 + cos^2x| + c | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
Latest revision as of 05:31, 5 September 2022
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} }
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\;=\; -\ \int \frac {du}{u}\;=\;-ln( 1 + u) + c\;=\;-| 1 + cos^2x| + c \end{align} }