A mash up of two functions

Wallis Theorem states that :

$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sin ^{ m }{ \theta } \cos ^{ n }{ \theta } d\theta } =\quad k\quad \frac { ((m-1)(m-3)(m-5)...........1\quad or\quad 2)((n-1)(n-3)(n-5)...........1\quad or\quad 2)\quad }{ ((m+n)(m+n-2)(m+n-4)..................1\quad or\quad 2) } \\ \\ where\quad m,n\quad are\quad whole\quad numbers\quad \\ \quad K=\quad \frac { \pi }{ 2 } \quad if\quad \quad m\quad \& \quad n\quad both\quad are\quad even.\\ \quad \quad =\quad 1\quad \quad \quad Otherwise.\\$ .

Let's Back to the Question

Put $x=\sin { \theta }$ .

$dx=\cos { \theta d\theta }$

So given Integral becomes:

$I=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sin ^{ 6 }{ \theta } \cos ^{ 2 }{ \theta } d\theta }$ .

By using Wallis Formula ( m=6 & n=2 )

$I=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sin ^{ 6 }{ \theta } \cos ^{ 2 }{ \theta } d\theta } \quad =\frac { \pi }{ 2 } \quad \frac { (5.3.1)(1) }{ (8.6.4.2) } =\frac { 5\pi }{ 256 }$ .

Clearly $I = \frac{1}{2} \beta(\frac{7}{2},\frac{3}{2})$ Where $\displaystyle \beta(m,n)$ is called $\beta$ function Also by using the properties of $\displaystyle \beta$ function we can write it in forms of $\displaystyle \Gamma$ function as $I= \frac{1}{2} \frac{\Gamma(\frac{7}{2}) \Gamma(\frac{3}{2})}{\Gamma(\frac{7}{2}+\frac{3}{2})}$ Hence $I=\frac{\frac{5}{2} \times \frac{3}{2} \times \frac{1}{2} \times \sqrt{\pi} \times \frac{1}{2} \times \sqrt{\pi}}{4 \times 3 \times 2 \times 1}$ After solving we get $\displaystyle I= \frac{5 \pi}{256} = \boxed{0.0613}$

Use beta function by taking x^2=u

a long and extended version that i used is by substituting $x= sin\theta$ then applying the property $f(x)=f(a+b-x)$ finally adding the two integrals as $2I=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { (sin\theta ) }^{ 2 }{ (cos\theta ) }^{ 2 }({ (sin\theta ) }^{ 4 }+{ (cos\theta ) }^{ 4 }) } d\theta$

after some simplification i got $2I=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { (sin2\theta ) }^{ 2 } }{ 4 } -\frac { { (sin2\theta ) }^{ 4 } }{ 8 } } d\theta$

hence after a long integration we finally get the answer $\boxed{\frac { 5\pi }{ 256 } }$

this is only calculus approached solution !!