`int (x^3 - 6x - 20)/(x + 5) dx` Find the indefinite integral.

Tuesday, March 31, 2015

$\displaystyle\int\frac{{{{x}}^{{3}}-{6}{x}-{20}}}{{{x}+{5}}}{\left.{d}{x}\right.}$ Find the indefinite integral.

$\displaystyle\int\frac{{{{x}}^{{3}}-{6}{x}-{20}}}{{{x}+{5}}}{\left.{d}{x}\right.}$

Let's evaluate the integral by applying integral substitution,

Let u=x+5, $\displaystyle\Rightarrow{x}={u}-{5}$

du=dx

$\displaystyle\int\frac{{{{x}}^{{3}}-{6}{x}-{20}}}{{{x}+{5}}}{\left.{d}{x}\right.}=\int\frac{{{{\left({u}-{5}\right)}}^{{3}}-{6}{\left({u}-{5}\right)}-{20}}}{{u}}{d}{u}$

$\displaystyle=\int\frac{{{\left({{u}}^{{3}}-{{5}}^{{3}}-{3}{{u}}^{{2}}\cdot{5}+{3}{u}\cdot{{5}}^{{2}}\right)}-{6}{u}+{30}-{20}}}{{u}}{d}{u}$

$\displaystyle=\int\frac{{{{u}}^{{3}}-{125}-{15}{{u}}^{{2}}+{75}{u}-{6}{u}+{10}}}{{u}}{d}{u}$

$\displaystyle=\int\frac{{{{u}}^{{3}}-{15}{{u}}^{{2}}+{69}{u}-{115}}}{{u}}{d}{u}$

$\displaystyle=\int{\left({{u}}^{{2}}-{15}{u}+{69}-\frac{{115}}{{u}}\right)}{d}{u}$

Now apply the sum rule,

$\displaystyle=\int{{u}}^{{2}}{d}{u}-\int{15}{u}{d}{u}-\int\frac{{115}}{{u}}{d}{u}+\int{69}{d}{u}$

$\displaystyle=\int{{u}}^{{2}}{d}{u}-\int{15}{u}{d}{u}-{115}\int\frac{{{d}{u}}}{{u}}+{69}\int{d}{u}$

Use the following common integrals,

$\displaystyle\int{{x}}^{{n}}{\left.{d}{x}\right.}=\frac{{{x}}^{{{n}+{1}}}}{{{n}+{1}}}$

and $\displaystyle\int\frac{{1}}{{x}}{\left.{d}{x}\right.}={\ln{{\left({\left|{x}\right|}\right)}}}$

$\displaystyle=\frac{{{u}}^{{3}}}{{3}}-{15}\frac{{{u}}^{{2}}}{{2}}-{115}{\ln}{\left|{u}\right|}+{69}{u}$

Substitute back u=x+5,

$\displaystyle=\frac{{{\left({x}+{5}\right)}}^{{3}}}{{3}}-\frac{{15}}{{2}}{{\left({x}+{5}\right)}}^{{2}}-{115}{\ln}{\left|{x}+{5}\right|}+{69}{\left({x}+{5}\right)}$

Add a constant C to the solution,

$\displaystyle=\frac{{{\left({x}+{5}\right)}}^{{3}}}{{3}}-\frac{{15}}{{2}}{{\left({x}+{5}\right)}}^{{2}}+{69}{\left({x}+{5}\right)}-{115}{\ln}{\left|{x}+{5}\right|}+{C}$

Ninenotes

Tuesday, March 31, 2015

$\displaystyle\int\frac{{{{x}}^{{3}}-{6}{x}-{20}}}{{{x}+{5}}}{\left.{d}{x}\right.}$ Find the indefinite integral.

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