Question

Plz give me answer ASAP​

Answer 1

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

$\rm :\longmapsto\:Let \: I \: = \displaystyle\int_0^{\dfrac{\pi}{2}}\tt \:x \: {sin}^{4}x \: {cos}^{6}x \: dx - - - (1)$

$\red{\rm :\longmapsto\:Change \: x \: \to \: \dfrac{\pi}{2} - x}$

$\rm :\longmapsto\ \: I = \displaystyle\int_0^{\dfrac{\pi}{2}}\tt \:\bigg(\dfrac{\pi}{2} - x\bigg){sin}^{4}\:\bigg(\dfrac{\pi}{2} - x\bigg){cos}^{6}\bigg(\dfrac{\pi}{2} - x\bigg)dx$

$\rm :\longmapsto\ \: I = \displaystyle\int_0^{\dfrac{\pi}{2}}\tt \:\bigg(\dfrac{\pi}{2} - x\bigg){cos}^{4}x \: {sin}^{6}x \: dx - - - (2)$

☆ On adding equation (1) and equation (2), we get

$\rm :\longmapsto\ \: 2I = \displaystyle\int_0^{\dfrac{\pi}{2}}\tt \:\bigg(\dfrac{\pi}{2}\bigg)({cos}^{4}x \: {sin}^{6}x + {sin}^{4}x {cos}^{6}x)\: dx$

$\rm :\longmapsto\ \: I = \displaystyle\int_0^{\dfrac{\pi}{2}}\tt \:\dfrac{\pi}{4}{cos}^{4}x \: {sin}^{4}x( {sin}^{2}x+{cos}^{2}x)\: dx$

$\rm :\longmapsto\ \: I = \displaystyle\int_0^{\dfrac{\pi}{2}}\tt \:\dfrac{\pi}{4}{cos}^{4}x \: {sin}^{4}x\: dx$

By using Walli's Formula, we get

$\rm :\longmapsto\:I \: = \dfrac{\pi}{4} \times \dfrac{(3 \times 1).(3 \times 1)}{(8 \times 6 \times 4 \times 2)} \times \dfrac{\pi}{2}$

$\bf :\longmapsto\:I \: = \dfrac{ {3\pi}^{2} }{1024}$

Additional Information :-

Walli's Formula :-

1. If n is odd natural number,

$\displaystyle\int_0^{\dfrac{\pi}{2}}\tt \: {sin}^{n}x = \dfrac{n - 1}{n} .\dfrac{n - 3}{n - 2} .\dfrac{n - 5}{n - 4} - - - \dfrac{2}{3}$

2. If n is odd natural number,

$\displaystyle\int_0^{\dfrac{\pi}{2}}\tt \: {cos}^{n}x = \dfrac{n - 1}{n} .\dfrac{n - 3}{n - 2} .\dfrac{n - 5}{n - 4} - - - \dfrac{2}{3}$

3. If n is even natural number,

$\displaystyle\int_0^{\dfrac{\pi}{2}}\tt \: {cos}^{n}x = \dfrac{n - 1}{n} .\dfrac{n - 3}{n - 2} .\dfrac{n - 5}{n - 4} - - - \dfrac{1}{2}\dfrac{\pi}{2}$

4. If n is even natural number,

$\displaystyle\int_0^{\dfrac{\pi}{2}}\tt \: {sin}^{n}x = \dfrac{n - 1}{n} .\dfrac{n - 3}{n - 2} .\dfrac{n - 5}{n - 4} - - - \dfrac{1}{2}\dfrac{\pi}{2}$

5. If m or n is odd natural number,

$\displaystyle\int_0^{\dfrac{\pi}{2}}\tt {sin}^{n}x {cos}^{m}x=\dfrac{(n - 1)(n - 3) - - - 2or1)(m - 1)(m - 3) - - - 2or1)}{(m + n)(m + n - 2)(m + n - 4)....or1}$

6. If m and n are even natural numbers,

$\displaystyle\int_0^{\dfrac{\pi}{2}}\tt {sin}^{n}x {cos}^{m}x=\dfrac{(n - 1)(n - 3) - - - 2or1)(m - 1)(m - 3) - - - 2)}{(m + n)(m + n - 2)(m + n - 4)....2}\dfrac{\pi}{2}$