∫sin^8 xdx ∫sin^6 xdx


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Question Number 36128 by mondodotto@gmail.com last updated on 29/May/18

$\int \sin^{8} x d x$ $\int \sin^{6} x d x$
Commented by Joel579 last updated on 29/May/18

Commented by abdo mathsup 649 cc last updated on 29/May/18
$∫ sin^8 xdx = ∫( ((1−cos(2x))/2))^4 dx = (1/(32)) ∫ (cos^2 (2x)−2cos(2x)+1)^2 dx = (1/(32))∫ ( ((1+cos(2x))/2) −2cos(2x) +1)^2 dx = (1/(64)) ∫ (1+cos(2x)−4cos(2x) +2)^2 dx =(1/(64)) ∫ (3 −3 cos(2x))^2 dx =(9/(64)) ∫ (cos^2 (2x) −2cos(2x) +1)dx =(9/(64))∫ ( ((1+cos(4x))/2) −2 cos(2x) +1)dx = (9/(128)) ∫ { 1+cos(4x) −4cos(2x) +2}dx =(9/(128))∫ { cos(4x) −4 cos(2x) +3}dx = (9/(4.128)) sin(4x) −((18)/(128))sin(2x) +((27)/(128)) +c$
$\int {s i n}^{8} x d x = \int {(\frac{1 - c o s (2 x)}{2})}^{4} d x$ $= \frac{1}{32} \int {({c o s}^{2} (2 x) - 2 c o s (2 x) + 1)}^{2} d x$ $= \frac{1}{32} \int {(\frac{1 + c o s (2 x)}{2} - 2 c o s (2 x) + 1)}^{2} d x$ $= \frac{1}{64} \int {(1 + c o s (2 x) - 4 c o s (2 x) + 2)}^{2} d x$ $= \frac{1}{64} \int {(3 - 3 c o s (2 x))}^{2} d x$ $= \frac{9}{64} \int ({c o s}^{2} (2 x) - 2 c o s (2 x) + 1) d x$ $= \frac{9}{64} \int (\frac{1 + c o s (4 x)}{2} - 2 c o s (2 x) + 1) d x$ $= \frac{9}{128} \int {1 + c o s (4 x) - 4 c o s (2 x) + 2} d x$ $= \frac{9}{128} \int {c o s (4 x) - 4 c o s (2 x) + 3} d x$ $= \frac{9}{4.128} s i n (4 x) - \frac{18}{128} s i n (2 x) + \frac{27}{128} + c$
Commented by abdo mathsup 649 cc last updated on 29/May/18

$\int {s i n}^{6} x d x = \int {(\frac{1 - c o s (2 x)}{3})}^{3} d x$ $= \frac{1}{27} \int {(1 - c o s (2 x))}^{2} (1 - c o s (2 x)) d x$ $= \frac{1}{27} \int (1 - 2 c o s (2 x) + {c o s}^{2} (2 x)) (1 - c o s (2 x)) d x$ $= \frac{1}{27} \int (1 - c o s (2 x) + \frac{1 + c o s (4 x)}{2}) (1 - c o s (2 x)) d x$ $= \frac{1}{54} \int (3 - 2 c o s (2 x) + c o s (4 x)) (1 - c o s (2 x)) d x$ $= \frac{1}{54} \int (3 - 2 c o s (2 x) + c o s (4 x)) d x$ $- \frac{1}{54} \int (3 - 2 c o s (2 x) + c o s (4 x)) c o s (2 x) d x$ $= \frac{3}{54} x - \frac{1}{54} s i n (2 x) + \frac{1}{216} s i n (4 x)$ $- \frac{3}{54} \int c o s (2 x) d x + \frac{2}{54} \int {c o s}^{2} (2 x) d x$ $- \frac{1}{54} \int c o s (2 x) c o s (4 x) d x$ $= . . . .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 29/May/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 29/May/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 29/May/18

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