∫sin51x(sinx)^(49) dx


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Question Number 47566 by tanmay.chaudhury50@gmail.com last updated on 11/Nov/18

$\int s i n 51 x {(s i n x)}^{49} d x$
	Answered by Smail last updated on 12/Nov/18

	$A = \int s i n (50 x + x) {s i n}^{49} (x) d x$ $= \int (s i n (50 x) c o s x + c o s (50 x) s i n (x)) {s i n}^{49} (x) d x$ $= \int (s i n (50 x) c o s (x) {s i n}^{49} (x) + c o s (50 x) {s i n}^{50} (x)) d x$ $= \int (s i n (50 x) {(\frac{1}{50} {s i n}^{50} x)}^{'} + {(\frac{1}{50} s i n (50 x))}^{'} {s i n}^{50} (x))$ $A = \frac{1}{50} s i n (50 x) {s i n}^{50} x$
	Answered by tanmay.chaudhury50@gmail.com last updated on 12/Nov/18
	$∫sin(50+1)x(sinx)^(49) dx ∫[sin50xcosx+cos50xsinx](sinx)^(49) dx ∫sin50xcosx(sinx)^(49) +cos50x(sinx)^(50) dx (1/(50))∫sin50x×50cosx(sinx)^(49) +50cos50x(sinx)^(50) dx (1/(50))∫[sin50x×(d/dx){(sinx)^(50) }+(sinx)^(50) ×(d/dx)(sin50x)]dx (1/(50))∫(d/dx){sin50x×(sinx)^(50) }dx (1/(50))×{sin50x×(sinx)^(50) }+c$
	$\int s i n (50 + 1) x {(s i n x)}^{49} d x$ $\int [s i n 50 x c o s x + c o s 50 x s i n x] {(s i n x)}^{49} d x$ $\int s i n 50 x c o s x {(s i n x)}^{49} + c o s 50 x {(s i n x)}^{50} d x$ $\frac{1}{50} \int s i n 50 x \times 50 c o s x {(s i n x)}^{49} + 50 c o s 50 x {(s i n x)}^{50} d x$ $\frac{1}{50} \int [s i n 50 x \times \frac{d}{d x} {{(s i n x)}^{50}} + {(s i n x)}^{50} \times \frac{d}{d x} (s i n 50 x)] d x$ $\frac{1}{50} \int \frac{d}{d x} {s i n 50 x \times {(s i n x)}^{50}} d x$ $\frac{1}{50} \times {s i n 50 x \times {(s i n x)}^{50}} + c$
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