New Notes Blog: `int e^x/(1-tan(e^x)) dx` Use integration tables to find the indefinite integral.

Tuesday, 3 June 2014

$\displaystyle\int\frac{{{e}}^{{x}}}{{{1}-{\tan{{\left({{e}}^{{x}}\right)}}}}}{\left.{d}{x}\right.}$ Use integration tables to find the indefinite integral.

$\displaystyle\int\frac{{{e}}^{{x}}}{{{1}-{\tan{{\left({{e}}^{{x}}\right)}}}}}{\left.{d}{x}\right.}$

First, let's apply integral substitution: $\displaystyle{u}={{e}}^{{x}}$

$\displaystyle\Rightarrow{d}{u}={{e}}^{{x}}{\left.{d}{x}\right.}$

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

Now from the integration tables:

$\displaystyle\int\frac{{1}}{{{1}\pm{\tan{{\left({u}\right)}}}}}{d}{u}=\frac{{1}}{{2}}{\left({u}\pm{\ln}{\left|{\cos{{\left({u}\right)}}}\pm{\sin{{\left({u}\right)}}}\right|}\right)}+{C}$

Using the above,

$\displaystyle=\frac{{1}}{{2}}{\left({u}-{\ln}{\left|{\cos{{\left({u}\right)}}}-{\sin{{\left({u}\right)}}}\right|}\right)}+{C}$

Substitute back $\displaystyle{u}={{e}}^{{x}}$

$\displaystyle=\frac{{1}}{{2}}{\left({{e}}^{{x}}-{\ln}{\left|{\cos{{\left({{e}}^{{x}}\right)}}}-{\sin{{\left({{e}}^{{x}}\right)}}}\right|}\right)}+{C}$

$\displaystyle\int\frac{{{e}}^{{x}}}{{{1}-{\tan{{\left({{e}}^{{x}}\right)}}}}}{\left.{d}{x}\right.}$

First, let's apply integral substitution: $\displaystyle{u}={{e}}^{{x}}$

$\displaystyle\Rightarrow{d}{u}={{e}}^{{x}}{\left.{d}{x}\right.}$

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

Now from the integration tables:

$\displaystyle\int\frac{{1}}{{{1}\pm{\tan{{\left({u}\right)}}}}}{d}{u}=\frac{{1}}{{2}}{\left({u}\pm{\ln}{\left|{\cos{{\left({u}\right)}}}\pm{\sin{{\left({u}\right)}}}\right|}\right)}+{C}$

Using the above,

$\displaystyle=\frac{{1}}{{2}}{\left({u}-{\ln}{\left|{\cos{{\left({u}\right)}}}-{\sin{{\left({u}\right)}}}\right|}\right)}+{C}$

Substitute back $\displaystyle{u}={{e}}^{{x}}$

$\displaystyle=\frac{{1}}{{2}}{\left({{e}}^{{x}}-{\ln}{\left|{\cos{{\left({{e}}^{{x}}\right)}}}-{\sin{{\left({{e}}^{{x}}\right)}}}\right|}\right)}+{C}$

New Notes Blog

Tuesday, 3 June 2014

$\displaystyle\int\frac{{{e}}^{{x}}}{{{1}-{\tan{{\left({{e}}^{{x}}\right)}}}}}{\left.{d}{x}\right.}$ Use integration tables to find the indefinite integral.

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Tuesday, 3 June 2014

Use integration tables to find the indefinite integral.

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Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle\int\frac{{{e}}^{{x}}}{{{1}-{\tan{{\left({{e}}^{{x}}\right)}}}}}{\left.{d}{x}\right.}$ Use integration tables to find the indefinite integral.

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?