`int sinh(1-2x) dx` Find the indefinite integral

Monday, February 28, 2011

$\displaystyle\int{\sinh{{\left({1}-{2}{x}\right)}}}{\left.{d}{x}\right.}$ Find the indefinite integral

It is known that $\displaystyle{\left({\cosh}\right)}'={\sinh{{x}}},$ so $\displaystyle\int{\sinh{{x}}}{\left.{d}{x}\right.}={\cosh{{x}}}+{C}.$ To reduce the given integral to the known one, make the substitution $\displaystyle{u}={1}-{2}{x}.$ Then $\displaystyle{d}{u}=-{2}{\left.{d}{x}\right.},$ $\displaystyle{\left.{d}{x}\right.}=-\frac{{1}}{{2}}{d}{u},$ and the integral becomes

$\displaystyle\int{\sinh{{\left({u}\right)}}}\cdot{\left(-\frac{{1}}{{2}}\right)}{d}{u}=-\frac{{1}}{{2}}\int{\sinh{{\left({u}\right)}}}{d}{u}=$

$\displaystyle=-\frac{{1}}{{2}}{\cosh{{\left({u}\right)}}}+{C}=-\frac{{1}}{{2}}{\cosh{{\left({1}-{2}{x}\right)}}}+{C},$

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

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Monday, February 28, 2011

$\displaystyle\int{\sinh{{\left({1}-{2}{x}\right)}}}{\left.{d}{x}\right.}$ 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?

Monday, February 28, 2011

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?

$\displaystyle\int{\sinh{{\left({1}-{2}{x}\right)}}}{\left.{d}{x}\right.}$ 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?