`int_0^1 (dx)/(x^2 + 1)^2` Evaluate the integral

Sunday, November 18, 2012

$\displaystyle{\int_{{0}}^{{1}}}\frac{{{\left.{d}{x}\right.}}}{{{\left({{x}}^{{2}}+{1}\right)}}^{{2}}}$ Evaluate the integral

First find the indefinite integral:

$\displaystyle\int\frac{{{\left.{d}{x}\right.}}}{{{\left({1}+{{x}}^{{2}}\right)}}^{{2}}}=\int\frac{{{1}+{{x}}^{{2}}-{{x}}^{{2}}}}{{{\left({1}+{{x}}^{{2}}\right)}}^{{2}}}{\left.{d}{x}\right.}=\int\frac{{{\left.{d}{x}\right.}}}{{{1}+{{x}}^{{2}}}}-\int\frac{{{{x}}^{{2}}{\left.{d}{x}\right.}}}{{{\left({1}+{{x}}^{{2}}\right)}}^{{2}}}.$

The first summand is $\displaystyle{\arctan{{\left({x}\right)}}},$ for the second apply integration by parts:

$\displaystyle{u}={x},$ $\displaystyle{d}{v}=\frac{{{x}{\left.{d}{x}\right.}}}{{{1}+{{x}}^{{2}}}},$ so $\displaystyle{d}{u}={\left.{d}{x}\right.}$ and $\displaystyle{v}=-\frac{{1}}{{2}}\frac{{1}}{{{1}+{{x}}^{{2}}}}.$

So $\displaystyle\int\frac{{{{x}}^{{2}}{\left.{d}{x}\right.}}}{{{\left({1}+{{x}}^{{2}}\right)}}^{{2}}}=-\frac{{1}}{{2}}\frac{{x}}{{{1}+{{x}}^{{2}}}}+\frac{{1}}{{2}}\int\frac{{{\left.{d}{x}\right.}}}{{{1}+{{x}}^{{2}}}}=-\frac{{1}}{{2}}\frac{{x}}{{{1}+{{x}}^{{2}}}}+\frac{{1}}{{2}}{\arctan{{\left({x}\right)}}}.$

Thus the indefinite integral is $\displaystyle\frac{{1}}{{2}}{\left(\frac{{x}}{{{1}+{{x}}^{{2}}}}+{\arctan{{\left({x}\right)}}}\right)}+{C},$

and the indefinite integral is $\displaystyle\frac{{1}}{{2}}{\left(\frac{{1}}{{2}}+\frac{\pi}{{4}}-{0}-{0}\right)}=\frac{{1}}{{4}}+\frac{\pi}{{8}}.$

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Sunday, November 18, 2012

$\displaystyle{\int_{{0}}^{{1}}}\frac{{{\left.{d}{x}\right.}}}{{{\left({{x}}^{{2}}+{1}\right)}}^{{2}}}$ Evaluate the 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?

Sunday, November 18, 2012

Evaluate the 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?

$\displaystyle{\int_{{0}}^{{1}}}\frac{{{\left.{d}{x}\right.}}}{{{\left({{x}}^{{2}}+{1}\right)}}^{{2}}}$ Evaluate the integral

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