Question

evaluate integral 0 to π sin^8x.cos^4 xdx​

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm \longrightarrow \: I = \displaystyle \int_{0}^{\pi} \rm \: {sin}^{8}x \: {cos}^{4}x \: dx$

$\rm :\longmapsto\:Let \: f(x) = {sin}^{8}x \: {cos}^{4}x$

Thus,

$\rm :\longmapsto\: \: f(\pi - x) = {sin}^{8}(\pi - x) \: {cos}^{4}(\pi - x)$

$\rm :\longmapsto\: \: f(\pi - x) = {sin}^{8}x \: {cos}^{4}x$

$\bf\implies \:f(\pi - x) = f(x)$

We know that,

$\begin{gathered}\begin{gathered}\bf\: \rm \longrightarrow \:\displaystyle \int_{0}^{a} \rm \: f(x) \: dx \: = \begin{cases} &\sf{ \displaystyle2\int_{0}^{\dfrac{a}{2} } \rm \: f(x) \: dx \: \: when \: f(a - x) = f(x)} \\ &\sf{0 \: \: when \: f(a - x) = - \: f(x)} \end{cases}\end{gathered}\end{gathered}$

So,

Given integral can be reduced to

$\rm \longrightarrow \: I = 2\displaystyle \int_{0}^{ \dfrac{\pi}{2} } \rm \: {sin}^{8}x \: {cos}^{4}x \: dx$

So, By using Walli's Formula,

$\rm \: = \: \: 2 \times \dfrac{(7.5.3.1)(3.1)}{12.10.8.6.4.2} \times \dfrac{\pi}{2}$

$\rm \: = \: \: \dfrac{7}{4.2.8.2.4.2} \times \pi$

$\rm \: = \: \: \dfrac{7\pi}{1024}$

Additional Information :-

Walli's Formula

1. If n is even natural number

$\displaystyle\int_{0}^{\dfrac{\pi}{2} } \rm \: {sin}^{n}x dx = \dfrac{(n - 1)(n - 3) - - - 1}{n(n - 2) - - - 2} \times \dfrac{\pi}{2}$

2. If n is even natural number

$\displaystyle\int_{0}^{\dfrac{\pi}{2} } \rm \: {cos}^{n}x dx = \dfrac{(n - 1)(n - 3) - - - 1}{n(n - 2) - - - 2} \times \dfrac{\pi}{2}$

3. If n is odd natural number

$\displaystyle\int_{0}^{\dfrac{\pi}{2} } \rm \: {sin}^{n}x dx = \dfrac{(n - 1)(n - 3) - - - 2}{n(n - 2) - - - 1}$

4. If n is odd natural number

$\displaystyle\int_{0}^{\dfrac{\pi}{2} } \rm \: {cos}^{n}x dx = \dfrac{(n - 1)(n - 3) - - - 2}{n(n - 2) - - - 1}$

5. If m and n both are even natural number

$\displaystyle\int_{0}^{\dfrac{\pi}{2} } \rm {sin}^{n}x {cos}^{m}x dx = \dfrac{ \{(n - 1)(n - 3) - - - 1 \} \{(m - 1)(m - 3) - - 1}{(m + n)((m + n - 2) - - - 2} \times \dfrac{\pi}{2}$

6. If m or n is odd natural number

$\displaystyle\int_{0}^{\dfrac{\pi}{2} } \rm {sin}^{n}x {cos}^{m}x dx = \dfrac{ \{(n - 1)(n - 3) - - - 2or1 \} \{(m - 1)(m - 3) - - 2or1}{(m + n)((m + n - 2) - - - 2or1}$