∫_( 0) ^(r π) sin^(2n) x dx =


Question and Answers Forum

All Questions Topic List
UNKNOWN Questions
Previous in All Question Next in All Question
Previous in UNKNOWN Next in UNKNOWN
Question Number 53694 by gunawan last updated on 25/Jan/19

$\underset{0}{\int^{r π}} \sin^{2 n} x d x =$
	Answered by tanmay.chaudhury50@gmail.com last updated on 25/Jan/19

	$\int_{0}^{π} s i n x d x =∣ - c o s x ∣_{0}^{π} = - (c o s π - c o s 0) = 2$ $s o t h e a r e a o f e a c h l o o p o f f (x) = s i n x i s 2$ $f (x) = {s i n}^{2 n} x$ $f (r π - x) = {[s i n (r π - x)]}^{2 n} = [e i t t h e r + s i n x o r - s i n x]$ $b u t {[s i n (r π - x)]}^{2 n} = {s i n}^{2 n} x$ $n o w \int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x w h e n f (2 a - x) = f (x)$ $s i n x = s i n (2 π + x) p e r i d o c i t y 2 π$ $\int_{0}^{r π} {s i n}^{2 n} x d x$ $= \int_{0}^{\frac{r}{2} \times 2 π} {s i n}^{2 n} x d x$ $= \frac{r}{2} \int_{0}^{2 π} {s i n}^{2 n} x d x$ $= \frac{r}{2} \times 2 \int_{0}^{π} {s i n}^{2 n} x d x [\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x w h e n f (2 a - x) = f (x)]$ $= 2 r \times \int_{0}^{\frac{π}{2}} {s i n}^{2 n} x d x$ $g a m m a f u n c t i o n . . .$ $2 \int_{0}^{\frac{π}{2}} {s i n}^{2 p - 1} {x c o s}^{2 q - 1} x d x = \frac{⌈ (p) ⌈ (q)}{⌈ (p + q)}$ $r \times 2 \int_{0}^{\frac{π}{2}} {s i n}^{2 n + 1 - 1} {x c o s}^{2 \times \frac{1}{2} - 1} d x$ $= r \times \frac{⌈ (2 n + 1) ⌈ (\frac{1}{2})}{⌈ (2 n + 1 + \frac{1}{2})}$ $i h a v e t r i e d t o s o l v e . . o t h e r s p l s c h e c k . . .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum