Help Notes: Evaluate the integral : `int

Thursday, 7 July 2011

The value of
$\displaystyle{\int_{{{0}}}^{{\frac{\pi}{{2}}}}}{{\cos}}^{{5}}{x}{\left.{d}{x}\right.}$ has to be
determined.

$\displaystyle{\int_{{{0}}}^{{\frac{\pi}{{2}}}}}{{\cos}}^{{5}}{x}$
dx

=> $\displaystyle{\int_{{{0}}}^{{\frac{\pi}{{2}}}}}{{\cos}}^{{4}}$
x*cos x dx

=> class="AM"> $\displaystyle{\int_{{{0}}}^{{\frac{\pi}{{2}}}}}{{\left({{\cos}}^{{2}}{x}\right)}}^{{2}}\cdot{\cos{{x}}}{\left.{d}{x}\right.}$

=> $\displaystyle{\int_{{{0}}}^{{\frac{\pi}{{2}}}}}{{\left({1}-{{\sin}}^{{2}}{x}\right)}}^{{2}}\cdot{\cos{{x}}}$
dx

=> $\displaystyle{\int_{{{0}}}^{{\frac{\pi}{{2}}}}}{\left({1}-\right.}$
2*sin^2x + sin^4x) *cos x dx

Let class="AM"> $\displaystyle{\sin{{x}}}={y}$

class="AM"> $\displaystyle{\left.{d}{y}\right.}={\cos{{x}}}\cdot{\left.{d}{x}\right.}$

class="AM"> $\displaystyle{\sin{{0}}}={0}$ and $\displaystyle{\sin{{\left(\frac{\pi}{{2}}\right)}}}$
= 1

The required integral is changed to:

$\displaystyle{\int_{{{0}}}^{{{1}}}}{1}-{2}\cdot{{y}}^{{2}}+{{y}}^{{4}}$
dy

=> $\displaystyle{y}-{2}\frac{{{y}}^{{3}}}{{3}}+$
y^5/5 between y = 0 and y = 1

=> class="AM"> $\displaystyle{1}-\frac{{2}}{{3}}+\frac{{1}}{{5}}$

=> class="AM"> $\displaystyle\frac{{8}}{{15}}$

The value of
$\displaystyle{\int_{{{0}}}^{{\frac{\pi}{{2}}}}}{{\cos}}^{{5}}{x}{\left.{d}{x}\right.}$ = class="AM"> $\displaystyle\frac{{8}}{{15}}$

Help Notes