7.1 Integration By Parts/54: Difference between revisions
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\int_{1}^{5} \left(x\ln(x) \right)dx = \frac{x^2\ln(x)}{2}\bigg|_{1}^{5} - \int_{1}^{5} \left(\frac{x^2}{2x} \right)dx = \frac{x^2\ln(x)}{2}\bigg|_{1}^{5} - \frac{1}{2}\int_{1}^{5} \left(x \right)dx = \frac{1\ln(1)}{2}-\frac{25\ln(5)}{2} -\left(\frac{1}{2}\right) \left( \frac{x^2}{2} \right) \bigg|_{1}^{5} = | \int_{1}^{5} \left(x\ln(x) \right)dx = \frac{x^2\ln(x)}{2}\bigg|_{1}^{5} - \int_{1}^{5} \left(\frac{x^2}{2x} \right)dx = \frac{x^2\ln(x)}{2}\bigg|_{1}^{5} - \frac{1}{2}\int_{1}^{5} \left(x \right)dx = \frac{1\ln(1)}{2}-\frac{25\ln(5)}{2} -\left(\frac{1}{2}\right) \left( \frac{x^2}{2} \right) \bigg|_{1}^{5} = 0-\frac{25}{2}\ln(5) -\frac{1}{2}\left(25-1\right) | ||
</math> | </math> | ||
Revision as of 04:08, 29 November 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 y=5\ln(x) , y=x\ln(x) }
![{\displaystyle {\begin{aligned}5\ln(x)&=x\ln(x)\\[1ex]&x=5\\[1ex]&x=1\\[1ex]5\ln(2)>2\ln(2)\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/cd1c3a218d1938f4fbde2e7b79218d92891c3f99)
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_{1}^{5}\left(5\ln(x) -x\ln(x) \right)dx = \int_{1}^{5} \left(5\ln(x) \right)dx - \int_{1}^{5} \left(x\ln(x) \right)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 \int_{1}^{5} \left(5\ln(x) \right)dx = 5 \int_{1}^{5} \left(\ln(x) \right)dx = 5\left(x\ln(x)\bigg|_{1}^{5}- \int_{1}^{5} \left(\frac{x}{x} \right)dx \right) = 5\left(x\ln(x) \bigg|_{1}^{5}- x \bigg|_{1}^{5} \right) = 5\left(5\ln(5)-1\ln(1) - \left(5-1 \right) \right) = 25\ln(5)-20 }
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 &= \ln(x) \quad dv= 1 dx \\ du &= \frac{1}{x} dx \quad v=x \\ \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 \int_{1}^{5} \left(x\ln(x) \right)dx = \frac{x^2\ln(x)}{2}\bigg|_{1}^{5} - \int_{1}^{5} \left(\frac{x^2}{2x} \right)dx = \frac{x^2\ln(x)}{2}\bigg|_{1}^{5} - \frac{1}{2}\int_{1}^{5} \left(x \right)dx = \frac{1\ln(1)}{2}-\frac{25\ln(5)}{2} -\left(\frac{1}{2}\right) \left( \frac{x^2}{2} \right) \bigg|_{1}^{5} = 0-\frac{25}{2}\ln(5) -\frac{1}{2}\left(25-1\right) }
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 &= \ln(x) \quad dv= x dx \\ du &= \frac{1}{x} \quad v=\frac{x^2}{2} \\ \end{align} }