`int cs ch(1/x)coth(1/x)/x^2 dx` Find the indefinite integral

Saturday, April 16, 2016

$\displaystyle\int{c}{s}{c}{h}{\left(\frac{{1}}{{x}}\right)}\frac{{\coth{{\left(\frac{{1}}{{x}}\right)}}}}{{{x}}^{{2}}}{\left.{d}{x}\right.}$ Find the indefinite integral

Those $\displaystyle\frac{{1}}{{x}}$ under the hyperbolic cosecant and cotangent are irritating, let's change them to more appropriate $\displaystyle{y}.$ Make the substitution $\displaystyle\frac{{1}}{{x}}={y},$ then $\displaystyle{x}=\frac{{1}}{{y}}$ and $\displaystyle{\left.{d}{x}\right.}=-\frac{{1}}{{{y}}^{{2}}}{\left.{d}{y}\right.}.$ The integral becomes

$\displaystyle-\int\frac{{{c}{s}{c}{h}{\left({y}\right)}{\coth{{\left({y}\right)}}}}}{{\frac{{1}}{{{y}}^{{2}}}}}\frac{{{\left.{d}{y}\right.}}}{{{y}}^{{2}}}=-\int{c}{s}{c}{h}{\left({y}\right)}{\coth{{\left({y}\right)}}}{\left.{d}{y}\right.}=$

|recall that $\displaystyle{c}{s}{c}{h}{\left({y}\right)}=\frac{{1}}{{\sinh{{\left({y}\right)}}}}$ and $\displaystyle{\coth{{\left({y}\right)}}}=\frac{{\cosh{{\left({y}\right)}}}}{{\sinh{{\left({y}\right)}}}}$ |

$\displaystyle=-\int\frac{{\cosh{{\left({y}\right)}}}}{{{{\sinh}}^{{2}}{\left({y}\right)}}}{\left.{d}{y}\right.}.$

The next substitution is $\displaystyle{u}={\sinh{{\left({y}\right)}}},$ then $\displaystyle{d}{u}={\cosh{{\left({y}\right)}}}{\left.{d}{y}\right.},$ and the integral becomes

$\displaystyle-\int\frac{{{d}{u}}}{{{u}}^{{2}}}=\frac{{1}}{{u}}+{C}=\frac{{1}}{{\sinh{{\left({y}\right)}}}}+{C}=\frac{{1}}{{\sinh{{\left(\frac{{1}}{{x}}\right)}}}}+{C}={c}{s}{c}{h}{\left(\frac{{1}}{{x}}\right)}+{C},$

where $\displaystyle{C}$ is an arbitrary constant.

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Saturday, April 16, 2016

$\displaystyle\int{c}{s}{c}{h}{\left(\frac{{1}}{{x}}\right)}\frac{{\coth{{\left(\frac{{1}}{{x}}\right)}}}}{{{x}}^{{2}}}{\left.{d}{x}\right.}$ Find the indefinite integral

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Saturday, April 16, 2016

Find the indefinite integral

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Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle\int{c}{s}{c}{h}{\left(\frac{{1}}{{x}}\right)}\frac{{\coth{{\left(\frac{{1}}{{x}}\right)}}}}{{{x}}^{{2}}}{\left.{d}{x}\right.}$ Find the indefinite integral

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?