New Notes Blog: `int 1 /(25+4x^2) dx` Find the indefinite integral

Saturday, 10 October 2015

$\displaystyle\int\frac{{1}}{{{25}+{4}{{x}}^{{2}}}}{\left.{d}{x}\right.}$ Find the indefinite integral

$\displaystyle\int\frac{{1}}{{{25}+{4}{{x}}^{{2}}}}{\left.{d}{x}\right.}$

Let's transform the denominator of the integral,

$\displaystyle\int\frac{{1}}{{{25}+{4}{{x}}^{{2}}}}{\left.{d}{x}\right.}=\int\frac{{1}}{{{4}{\left({{x}}^{{2}}+\frac{{25}}{{4}}\right)}}}{\left.{d}{x}\right.}$

Take the constant out,

$\displaystyle=\frac{{1}}{{4}}\int\frac{{1}}{{{{x}}^{{2}}+{{\left(\frac{{5}}{{2}}\right)}}^{{2}}}}{\left.{d}{x}\right.}$

Now use the standard integral: $\displaystyle\int\frac{{1}}{{{{x}}^{{2}}+{{a}}^{{2}}}}{\left.{d}{x}\right.}=\frac{{1}}{{a}}{\arctan{{\left(\frac{{x}}{{a}}\right)}}}$

$\displaystyle=\frac{{1}}{{4}}{\left(\frac{{1}}{{\frac{{5}}{{2}}}}\right)}{\arctan{{\left(\frac{{x}}{{\frac{{5}}{{2}}}}\right)}}}$

simplify and add a constant C to the solution,

$\displaystyle={\left(\frac{{1}}{{4}}\right)}{\left(\frac{{2}}{{5}}\right)}{\arctan{{\left(\frac{{{2}{x}}}{{5}}\right)}}}+{C}$

$\displaystyle=\frac{{1}}{{10}}{\arctan{{\left(\frac{{{2}{x}}}{{5}}\right)}}}+{C}$

$\displaystyle\int\frac{{1}}{{{25}+{4}{{x}}^{{2}}}}{\left.{d}{x}\right.}$

Let's transform the denominator of the integral,

$\displaystyle\int\frac{{1}}{{{25}+{4}{{x}}^{{2}}}}{\left.{d}{x}\right.}=\int\frac{{1}}{{{4}{\left({{x}}^{{2}}+\frac{{25}}{{4}}\right)}}}{\left.{d}{x}\right.}$

Take the constant out,

$\displaystyle=\frac{{1}}{{4}}\int\frac{{1}}{{{{x}}^{{2}}+{{\left(\frac{{5}}{{2}}\right)}}^{{2}}}}{\left.{d}{x}\right.}$

Now use the standard integral: $\displaystyle\int\frac{{1}}{{{{x}}^{{2}}+{{a}}^{{2}}}}{\left.{d}{x}\right.}=\frac{{1}}{{a}}{\arctan{{\left(\frac{{x}}{{a}}\right)}}}$

$\displaystyle=\frac{{1}}{{4}}{\left(\frac{{1}}{{\frac{{5}}{{2}}}}\right)}{\arctan{{\left(\frac{{x}}{{\frac{{5}}{{2}}}}\right)}}}$

simplify and add a constant C to the solution,

$\displaystyle={\left(\frac{{1}}{{4}}\right)}{\left(\frac{{2}}{{5}}\right)}{\arctan{{\left(\frac{{{2}{x}}}{{5}}\right)}}}+{C}$

$\displaystyle=\frac{{1}}{{10}}{\arctan{{\left(\frac{{{2}{x}}}{{5}}\right)}}}+{C}$

New Notes Blog

Saturday, 10 October 2015

$\displaystyle\int\frac{{1}}{{{25}+{4}{{x}}^{{2}}}}{\left.{d}{x}\right.}$ 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?

Saturday, 10 October 2015

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?

$\displaystyle\int\frac{{1}}{{{25}+{4}{{x}}^{{2}}}}{\left.{d}{x}\right.}$ 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?