Question

integral of sin2x/1+sin^2x​

Answer 1

$\implies \displaystyle\int \:\sf \dfrac{sin \: 2x}{ {sin}^{2}x }dx$

We know that :-

$\sf sin \: 2t \: = 2 \: sin \: t \: cos\:t$

$\implies \displaystyle\int \sf\dfrac{2sin \: x \: cos \: x}{ {sin}^{2}x } dx$

$\implies 2 \displaystyle\int \:\sf \dfrac{ \: cos \: x}{ {sin} \: x } dx$

$\sf \: Let \: sin \: x \: = m \\ \sf \therefore \: cos \: x \: dx = dm$

$\implies 2\displaystyle\int \:\sf \dfrac{ dm}{ m}$

$\implies 2 \sf \: log (m) + c$

Where c is constant.

Put m = sin x

$\implies \sf \:2 log (sin \: x) + c$

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Learn more :-

∫ 1 dx = x + C

∫ sin x dx = – cos x + C

∫ cos x dx = sin x + C

∫ sec²x dx = tan x + C

∫ csc²x dx = -cot x + C

∫ sec x (tan x) dx = sec x + C

∫ csc x ( cot x) dx = – csc x + C

∫ (1/x) dx = ln |x| + C

∫ ex dx = ex+ C

∫ ax dx = (ax/ln a) + C