Integral of sin2x/1+sin^2x

1.	Integral of sin2x/1+sin^2x
Answer» $\implies \displaystyle\int \:\sf \dfrac{sin \: 2x}{ {sin}^{2}x }dx$ We know that :- $\sf sin \: 2t \: = 2 \: sin \: t \: cos\:t$ $\implies \displaystyle\int \sf\dfrac{2sin \: x \: cos \: x}{ {sin}^{2}x } dx$ $\implies 2 \displaystyle\int \:\sf \dfrac{ \: cos \: x}{ {sin} \: x } dx$ $\sf \: Let \: sin \: x \: = m \\ \sf \therefore \: cos \: x \: dx = dm$ $\implies 2\displaystyle\int \:\sf \dfrac{ dm}{ m}$ $\implies 2 \sf \: log (m) + <klux>C</klux>$ Where c is constant. Put m = sin x $\implies \sf \:2 log (sin \: x) + c$ __________________ 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

Answer»

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