The Notes: `int (sin^3(sqrt(x)))/sqrt(x) dx` Evaluate the integral

Monday, July 25, 2016

$\displaystyle\int\frac{{{{\sin}}^{{3}}{\left(\sqrt{{{x}}}\right)}}}{\sqrt{{{x}}}}{\left.{d}{x}\right.}$ Evaluate the integral

$\displaystyle\int{\left[\frac{{{{\sin}}^{{3}}\sqrt{{x}}}}{{\sqrt{{x}}}}\right]}{\left.{d}{x}\right.}=$

Integrate using the u-substitution method. For this problem the u-substitution method will be used twice. The first time we substitute, let's use the variable y.

Let

$\displaystyle{y}=\sqrt{{x}}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{1}}{{{2}\sqrt{{x}}}}$

$\displaystyle{\left.{d}{x}\right.}={2}\sqrt{{x}}{\left.{d}{y}\right.}$

$\displaystyle\int{\left[\frac{{{{\sin}}^{{3}}{\left({y}\right)}}}{{y}}\cdot{2}\sqrt{{x}}{\left.{d}{y}\right.}=\right.}$

$\displaystyle{2}\int{\left[\frac{{{{\sin}}^{{3}}{\left({y}\right)}}}{{{y}}}\right]}\cdot{y}{\left.{d}{y}\right.}=$

$\displaystyle{2}\int{{\sin}}^{{3}}{\left({y}\right)}{\left.{d}{y}\right.}=$

$\displaystyle{2}\int{{\sin}}^{{2}}{\left({y}\right)}{\sin{{\left({y}\right)}}}{\left.{d}{y}\right.}=$

$\displaystyle{2}\int{\left({1}-{{\cos}}^{{2}}{\left({y}\right)}\right)}{\sin{{\left({y}\right)}}}{\left.{d}{y}\right.}=$

The u-substitution method will be used a second time. We will use the variable u.

Let

$\displaystyle{u}={\cos{{y}}}$

$\displaystyle\frac{{{d}{u}}}{{\left.{d}{y}}}=-{\sin{{\left({y}\right)}}}$

$\displaystyle{\left.{d}{y}\right.}=\frac{{-{\sin{{\left({y}\right)}}}}}{{{d}{u}}}$

$\displaystyle{2}\int{\left({1}-{{u}}^{{2}}\right)}{\sin{{\left({y}\right)}}}{\left[\frac{{{d}{u}}}{{-{\sin{{\left({y}\right)}}}}}\right]}=$

$\displaystyle-{2}\int{\left({1}-{{u}}^{{2}}\right)}{d}{u}=$

$\displaystyle-{2}{\left[{u}-\frac{{1}}{{3}}{{u}}^{{3}}\right]}+{C}=$

$\displaystyle-{2}{u}+\frac{{2}}{{3}}{{u}}^{{3}}+{C}$

Substitute in for u. $\displaystyle{u}={\cos{{\left({y}\right)}}}$

$\displaystyle-{2}{\cos{{\left({y}\right)}}}+\frac{{2}}{{3}}{{\cos}}^{{3}}{\left({y}\right)}+{C}$

Substitute in for y. $\displaystyle{y}=\sqrt{{x}}$

$\displaystyle-{2}{\cos{{\left(\sqrt{{x}}\right)}}}+\frac{{2}}{{3}}{{\cos}}^{{3}}{\left(\sqrt{{x}}\right)}+{C}$

The final answer is: $\displaystyle-{2}{\cos{{\left(\sqrt{{x}}\right)}}}+\frac{{2}}{{3}}{{\cos}}^{{3}}{\left(\sqrt{{x}}\right)}+{C}$

The Notes

Monday, July 25, 2016

$\displaystyle\int\frac{{{{\sin}}^{{3}}{\left(\sqrt{{{x}}}\right)}}}{\sqrt{{{x}}}}{\left.{d}{x}\right.}$ Evaluate the integral

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