The Notes: `int_0^(pi/2) sin^5(x) dx` Evaluate the integral

Tuesday, November 26, 2013

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{2}}}}}{{\sin}}^{{5}}{\left({x}\right)}{\left.{d}{x}\right.}$ Evaluate the integral

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{2}}}}}{{\sin}}^{{5}}{\left({x}\right)}{\left.{d}{x}\right.}$

Let's first evaluate the definite integral,by rewriting the integrand as,

$\displaystyle=\int{{\sin}}^{{4}}{\left({x}\right)}{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}$

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

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

Now apply the integral substitution,

Let $\displaystyle{u}={\cos{{\left({x}\right)}}}$

$\displaystyle\Rightarrow{d}{u}=-{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}$

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

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

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

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

Substitute back $\displaystyle{u}={\cos{{\left({x}\right)}}}$

and adding a constant C to the solution,

$\displaystyle\int{{\sin}}^{{5}}{\left({x}\right)}{\left.{d}{x}\right.}=-{\cos{{\left({x}\right)}}}+\frac{{2}}{{3}}{{\cos}}^{{3}}{\left({x}\right)}-\frac{{1}}{{5}}{{\cos}}^{{5}}{\left({x}\right)}+{C}$

Now let's evaluate the definite integral,

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{2}}}}}{{\sin}}^{{5}}{\left({x}\right)}{\left.{d}{x}\right.}={{\left[-{\cos{{\left({x}\right)}}}+\frac{{2}}{{3}}{{\cos}}^{{3}}{\left({x}\right)}-\frac{{1}}{{5}}{{\cos}}^{{5}}{\left({x}\right)}\right]}_{{0}}^{{\frac{\pi}{{2}}}}}$

$\displaystyle={\left[-{\cos{{\left(\frac{\pi}{{2}}\right)}}}+\frac{{2}}{{3}}{{\cos}}^{{3}}{\left(\frac{\pi}{{2}}\right)}-\frac{{1}}{{5}}{{\cos}}^{{5}}{\left(\frac{\pi}{{2}}\right)}\right]}-{\left[-{\cos{{\left({0}\right)}}}+\frac{{2}}{{3}}{{\cos}}^{{3}}{\left({0}\right)}-\frac{{1}}{{5}}{{\cos}}^{{5}}{\left({0}\right)}\right]}$