Question

What is the value of
$\displaystyle \dfrac{\int_0^{\frac{\pi}{2}}\sin^{102}x \ dx }{\int_0^{\frac{\pi}{2}}\sin^{100}x \ dx }$

Answer 1

Reduction Formula is attached above ...

$\dfrac{\int_0^{\frac{\pi}{2}}\sin^{102}x \ dx }{\int_0^{\frac{\pi}{2}}\sin^{100}x \ dx } \\ \frac{ \frac{101 \times 99 \times 97.....1}{102 !} }{ \frac{99 \times 97 \times 95...1}{100 ! } } \\ \frac{ \frac{101 \times 99}{102 \times 101} }{ \frac{1}{1} } \\ \frac{99}{102} \\ \frac{33}{34}$

Answer 2

EXPLANATION.

$\sf \implies \dfrac{\displaystyle\int\limits^\frac{\pi}{2} _0 {sin^{102} (}x )\, dx }{\displaystyle\int\limits^\frac{\pi}{2} _0 {sin^{100} (x)} \, dx }$

As we know that,

In this type of questions we can apply,

WALLIS THEOREAM.

This method is only apply when equation is integrate from 0 to π/2.

$\sf \implies \displaystyle\int\limits^\frac{\pi}{2} _0 {sin^{m}x .cos^{n} x } \, dx = \dfrac{\bigg[(n - 1)(n - 3),,,1 or 2\bigg]\bigg[(m - 1)(m - 3),,,,,1 or 2\bigg]}{(m + n)(m + n - 2),,,,1 or 2.} \times K$

m, n are not negative integers.

where,

k = π/2 if m, n both are even.

k = 1 (otherwise ).

$\sf \implies \dfrac{\displaystyle\int\limits^\frac{\pi}{2} _0 {sin^{2}(x) sin^{100}(x) } \, dx }{\displaystyle\int\limits^\frac{\pi}{2} _0 {sin^{100} (x)} \, dx }$

Using reduction formula in equation, we get.

$\sf \implies \dfrac{\dfrac{\dfrac{(2 - 1)\bigg[(100 - 1)(100 - 3)(100 - 5),,,,1\bigg]}{(2)\bigg[(100)(100 - 2)(100 - 4),,,,,2\bigg]} \times \dfrac{\pi}{2} \times \dfrac{\pi}{2} }{\bigg[(100 - 1)(100 - 3)(100 - 5),,,,1\bigg]} }{\bigg[(100)(100 - 2)(100 - 4),,,,2\bigg]} \times \dfrac{\pi}{2}$

$\sf \implies \dfrac{\dfrac{1(99)(97)(95),,,,1}{(2)(100)(98)(96),,,,,,2} \times \dfrac{\pi}{2} }{\dfrac{(99)(97)(96),,,,1}{(100)(98)(96),,,,,2} }$

$\sf \implies \dfrac{99 \times 97 \times 95,,,,1}{2 \times 100 \times 98 \times 96 ,,,2} \times \dfrac{\pi}{2} \times \dfrac{100 \times 98 \times 96 ,,,,2 }{99 \times 97 \times 95,,,,,1}$

$\sf \implies \dfrac{\pi}{4}$

$\sf \implies \dfrac{\displaystyle\int\limits^\frac{\pi}{2} _0 {sin^{102} (}x )\, dx }{\displaystyle\int\limits^\frac{\pi}{2} _0 {sin^{100} (x)} \, dx } = \dfrac{\pi}{4}$