Blog: `int sinh^2(x) cosh(x) dx` Evaluate the indefinite integral.

Monday, December 29, 2014

$\displaystyle\int{{\sinh}}^{{2}}{\left({x}\right)}{\cosh{{\left({x}\right)}}}{\left.{d}{x}\right.}$ Evaluate the indefinite integral.

You need to evaluate the indefinite integral by using the substitution $\displaystyle{\sinh{{x}}}={t}$ , such that:

$\displaystyle{\sinh{{x}}}={t}\Rightarrow{\cosh{{x}}}{\left.{d}{x}\right.}={\left.{d}{t}\right.}$

$\displaystyle\int{{\sinh}}^{{2}}{x}\cdot{\cosh{{x}}}{\left.{d}{x}\right.}=\int{{t}}^{{2}}{\left.{d}{t}\right.}$

Using the formula $\displaystyle\int{{t}}^{{n}}{\left.{d}{t}\right.}=\frac{{{{t}}^{{{n}+{1}}}}}{{{n}+{1}}}$ yields:

$\displaystyle\int{{t}}^{{2}}{\left.{d}{t}\right.}=\frac{{{t}}^{{3}}}{{3}}+{c}$

Replacing back $\displaystyle{\sinh{{x}}}$ for t yields:

$\displaystyle\int{{\sinh}}^{{2}}{x}\cdot{\cosh{{x}}}{\left.{d}{x}\right.}=\frac{{{{\sinh}}^{{3}}{x}}}{{3}}+{c}$

Hence, evaluating the indefinite integral, yields $\displaystyle\int{{\sinh}}^{{2}}{x}\cdot{\cosh{{x}}}{\left.{d}{x}\right.}=\frac{{{{\sinh}}^{{3}}{x}}}{{3}}+{c}.$

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Monday, December 29, 2014

$\displaystyle\int{{\sinh}}^{{2}}{\left({x}\right)}{\cosh{{\left({x}\right)}}}{\left.{d}{x}\right.}$ Evaluate the indefinite integral.

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