Blog: `int y e^(0.2y) dy` Evaluate the integral

Monday, March 3, 2014

$\displaystyle\int{y}{{e}}^{{{0.2}{y}}}{\left.{d}{y}\right.}$ Evaluate the integral

You need to use the substitution $\displaystyle{0.2}{y}={u}$ , such that:

$\displaystyle{0.2}{y}={u}\Rightarrow{0.2}{\left.{d}{y}\right.}={d}{u}\Rightarrow{\left.{d}{y}\right.}=\frac{{{d}{u}}}{{{0.2}}}$

Replacing the variable, yields:

$\displaystyle\int{y}\cdot{{e}}^{{{0.2}{y}}}{\left.{d}{y}\right.}={\left(\frac{{10}}{{2}}\right)}\int\frac{{u}}{{{0.2}}}\cdot{{e}}^{{u}}{d}{u}$

You need to use the integration by parts such that:

$\displaystyle\int{f{{d}}}{g{=}}{f{{g{-}}}}\int{g{{d}}}{f{}}$

$\displaystyle{f{=}}{u}\Rightarrow{d}{f{=}}{d}{u}$

$\displaystyle{d}{g{=}}{{e}}^{{u}}\Rightarrow{g{=}}{{e}}^{{u}}$

2 $\displaystyle{5}\int{u}\cdot{{e}}^{{u}}{d}{u}={25}{\left({u}\cdot{{e}}^{{u}}-\int{{e}}^{{u}}{d}{u}\right)}$

$\displaystyle{25}\int{u}\cdot{{e}}^{{u}}{d}{u}={25}{u}\cdot{{e}}^{{u}}-{25}{{e}}^{{u}}+{c}$

Replacing back the variable, yields:

$\displaystyle\int{y}\cdot{{e}}^{{{0.2}}}{\left.{d}{y}\right.}={25}{\left({\left({0.2}{y}\right)}\cdot{{e}}^{{{0.2}{y}}}-{{e}}^{{{0.2}{y}}}\right)}+{c}$

Hence, evaluating the integral, using substitution, then integration by parts, yields $\displaystyle\int{y}\cdot{{e}}^{{{0.2}}}{\left.{d}{y}\right.}={\left(\frac{{{{e}}^{{{0.2}{y}}}}}{{9}}\right)}{\left({0.2}{y}-{1}\right)}+{c}$

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Monday, March 3, 2014

$\displaystyle\int{y}{{e}}^{{{0.2}{y}}}{\left.{d}{y}\right.}$ Evaluate the integral

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What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?

Monday, March 3, 2014

Evaluate the integral

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What is the Exposition, Rising Action, Climax, and Falling Action of &quot;One Thousand Dollars&quot;?

$\displaystyle\int{y}{{e}}^{{{0.2}{y}}}{\left.{d}{y}\right.}$ Evaluate the integral

What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?