Find -0-2sin 4 x d xif In-0-2sin n x d x-n-1-nn-3-n-2n-5-n-4- I0

∫ _0^π/2sin ^n x dx=n 1nn 3n 2.....π2 ∫ _0^πsin ^4 xdx =34 12π2=3π16 ...

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Integration by Parts

Find ∫0π/2si...

Question

Find $\int_{0}^{\frac{π}{2}} {sin}^{4} x d x$
if $I (n) = \int_{0}^{\frac{π}{2}} {sin}^{n} x d x = \frac{(n - 1)}{n} \frac{(n - 3)}{(n - 2)} \frac{(n - 5)}{(n - 4)} \dots I_{0}$

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Solution

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Similar questions

\int_{0}^{π / 2} {sin}^{6} x d x = \frac{5 π}{16}

\int_{0}^{π / 2} {sin}^{n} x d x

\frac{n - 1}{n} \frac{n - 3}{n - 2} \frac{n - 5}{n - 4} . . . \frac{1}{2} \times \frac{π}{2}

I = \int_{0}^{π / 2} {cos}^{n} x {sin}^{n} x dx = k \int_{0}^{π / 2} {sin}^{n} x dx

6

n - \frac{n (n - 1) (n - 2)}{⌊ 3} \cdot 3 + \frac{n (n - 1) (n - 2) (n - 3) (n - 4)}{⌊ 5} \cdot 3^{2} - . . . . .,

n - \frac{n (n - 1) (n - 2)}{⌊ 3} \cdot \frac{1}{3} + \frac{n (n - 1) (n - 2) (n - 3) (n - 4)}{⌊ 5} \cdot \frac{1}{3^{2}} - . . . . .,

∵ \int_{0}^{\frac{π}{2}} {sin}^{m} {x c o s}^{n} x d x = \frac{n - 1}{m + n} \frac{n - 3}{m + n - 2} . . . . \frac{1}{m + 2} \frac{m - 1}{m} \frac{m - 3}{m - 2} . . . . . \frac{1}{2} \frac{π}{2}

$n^{n} - n (n - 1)^{n} + \frac{n (n - 1)}{2!} (n - 2)^{n} - . . . \infty = \frac{3}{a} \times n!$ .

Find $a$

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