5.5 The Substitution Rule/2: Difference between revisions
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\int x^3(2+x^4)^5dx &= \int (x^ | \int x^3(2+x^4)^5dx &= \int (x^3dx)(2+x^4) = \int \left(\frac{1}{4}du\right)(u) = \frac{1}{4}\int u\,du \\[2ex] | ||
&= \frac{1}{4} \left[\frac{u^{1+1}}{1+1}\right] + C = \frac{u^2}{8} + C \\[2ex] | &= \frac{1}{4} \left[\frac{u^{1+1}}{1+1}\right] + C = \frac{u^2}{8} + C \\[2ex] | ||
Revision as of 19:00, 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 x^3(2+x^4)^5dx \text{,} \quad u=2+x^4 }
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 &=2+x^4 \\[2ex] du &= 4x^3dx \\[2ex] \frac{1}{4}du &= x^3dx \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 x^3(2+x^4)^5dx &= \int (x^3dx)(2+x^4) = \int \left(\frac{1}{4}du\right)(u) = \frac{1}{4}\int u\,du \\[2ex] &= \frac{1}{4} \left[\frac{u^{1+1}}{1+1}\right] + C = \frac{u^2}{8} + C \\[2ex] &= \frac{(2+x^4)^2}{8} + C \end{align} }