5.5 The Substitution Rule/5: Difference between revisions
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\int \cos^{3}{(\theta)}\sin{(\theta)}d{\theta)} \text{,} \quad u=\cos{(\theta)} | \begin{align} | ||
\int \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} \text{,} \quad u=\cos{(\theta)}\\[2ex] | |||
\end{align} | |||
</math> | </math> | ||
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\end{align} | \end{align} | ||
</math> | </math> | ||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
\int \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} = \ | \int \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} &= -\int u^{3}du \\[2ex] | ||
&= \frac{-u^{4}}{4} + C = \frac{-\cos^{4}{(\theta)}}{4} + C | &= \frac{-u^{4}}{4} + C = \frac{-\cos^{4}{(\theta)}}{4} + C \\[2ex] | ||
&= \frac{-1}{4}\cos^{4}{(\theta)} + C | &= \frac{-1}{4}\cos^{4}{(\theta)} + C | ||
Latest revision as of 19:52, 22 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 \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} \text{,} \quad u=\cos{(\theta)}\\[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 \begin{align} u &=\cos{(\theta)} \\[2ex] du &=-\sin{(\theta)}d{(\theta)} \\[2ex] -du &=\sin{(\theta)}d{(\theta)} \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 \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} &= -\int u^{3}du \\[2ex] &= \frac{-u^{4}}{4} + C = \frac{-\cos^{4}{(\theta)}}{4} + C \\[2ex] &= \frac{-1}{4}\cos^{4}{(\theta)} + C \end{align} }