The Notes: `int cos(theta)cos^5(sin(theta)) d theta` Evaluate the integral

Thursday, September 25, 2014

$\displaystyle\int{\cos{{\left(\theta\right)}}}{{\cos}}^{{5}}{\left({\sin{{\left(\theta\right)}}}\right)}{d}\theta$ Evaluate the integral

$\displaystyle\int{\cos{{\left(\theta\right)}}}{{\cos}}^{{5}}{\left({\sin{{\left(\theta\right)}}}\right)}{d}\theta$

Apply integral substitution,

Let $\displaystyle{u}={\sin{{\left(\theta\right)}}}$

$\displaystyle\Rightarrow{d}{u}={\cos{{\left(\theta\right)}}}{d}\theta$

$\displaystyle=\int{{\cos}}^{{5}}{\left({u}\right)}{d}{u}$

Rewrite the above integrand as,

$\displaystyle=\int{{\cos}}^{{4}}{\left({u}\right)}{\cos{{\left({u}\right)}}}{d}{u}$

Now use the identity: $\displaystyle{{\cos}}^{{2}}{\left({x}\right)}={1}-{{\sin}}^{{2}}{\left({x}\right)}$

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

Apply the integral substitution,

Let $\displaystyle{v}={\sin{{\left({u}\right)}}}$

$\displaystyle\Rightarrow{d}{v}={\cos{{\left({u}\right)}}}{d}{u}$

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

$\displaystyle=\int{\left({1}-{2}{{v}}^{{2}}+{{v}}^{{4}}\right)}{d}{v}$

$\displaystyle=\int{1}{d}{v}-{2}\int{{v}}^{{2}}{d}{v}+\int{{v}}^{{4}}{d}{v}$

$\displaystyle={v}-{2}{\left(\frac{{{v}}^{{3}}}{{3}}\right)}+\frac{{{v}}^{{5}}}{{5}}$

Substitute back $\displaystyle{v}={\sin{{\left({u}\right)}}}$

$\displaystyle={\sin{{\left({u}\right)}}}-\frac{{2}}{{3}}{{\sin}}^{{3}}{\left({u}\right)}+\frac{{1}}{{5}}{{\sin}}^{{5}}{\left({u}\right)}$

Substitute back $\displaystyle{u}={\sin{{\left(\theta\right)}}}$

$\displaystyle={\sin{{\left({\sin{{\left(\theta\right)}}}\right)}}}-\frac{{2}}{{3}}{{\sin}}^{{3}}{\left({\sin{{\left(\theta\right)}}}\right)}+\frac{{1}}{{5}}{{\sin}}^{{5}}{\left({\sin{{\left(\theta\right)}}}\right)}$

Add a constant C to the solution,

The Notes

Thursday, September 25, 2014

$\displaystyle\int{\cos{{\left(\theta\right)}}}{{\cos}}^{{5}}{\left({\sin{{\left(\theta\right)}}}\right)}{d}\theta$ Evaluate the integral

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