5.5 The Substitution Rule/54: Difference between revisions
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\int_{0}^{\sqrt{\pi}} x\cos{(x^2)}\,dx &= \int_{0}^{\sqrt{\pi}} (xdx)\cos{(x^2)} \\[2ex] | \int_{0}^{\sqrt{\pi}} x\cos{(x^2)}\,dx &= \int_{0}^{\sqrt{\pi}} (xdx)\cos{(x^2)} \\[2ex] | ||
&= \int_{0}^{\pi} (\frac{1}{2}du)\cos{(u)} = \frac{1}{2}\int_{0}^{\pi} \cos{(u)}du \\[2ex] | &= \int_{0}^{\pi} \left(\frac{1}{2}du\right)\cos{(u)} = \frac{1}{2}\int_{0}^{\pi} \cos{(u)}du \\[2ex] | ||
&= \frac{1}{2}\sin{(u)}\bigg|_{0}^{\pi} \\[2ex] | &= \left[\frac{1}{2}\right]\sin{(u)}\bigg|_{0}^{\pi} \\[2ex] | ||
&= \frac{1}{2}\sin{((\pi))} - \frac{1}{2}\sin{((0))} \\[2ex] | &= \frac{1}{2}\sin{((\pi))} - \frac{1}{2}\sin{((0))} \\[2ex] | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
Revision as of 19:27, 26 August 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 \int_{0}^{\sqrt{\pi}} x\cos{(x^2)}\,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} u &=x^2 \\[2ex] du &= 2xdx \\[2ex] \frac{1}{2}du &= xdx \\[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} \int_{0}^{\sqrt{\pi}} x\cos{(x^2)}\,dx &= \int_{0}^{\sqrt{\pi}} (xdx)\cos{(x^2)} \\[2ex] &= \int_{0}^{\pi} \left(\frac{1}{2}du\right)\cos{(u)} = \frac{1}{2}\int_{0}^{\pi} \cos{(u)}du \\[2ex] &= \left[\frac{1}{2}\right]\sin{(u)}\bigg|_{0}^{\pi} \\[2ex] &= \frac{1}{2}\sin{((\pi))} - \frac{1}{2}\sin{((0))} \\[2ex] \end{align} }