∫((cscx)/(cos(2x)+2cos^2 x))dx pleas sir help me


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Question Number 92421 by mhmd last updated on 06/May/20

$\int \frac{c s c x}{c o s (2 x) + 2 {c o s}^{2} x} d x$ $p l e a s s i r h e l p m e$
	Answered by niroj last updated on 07/May/20
	$∫ (( cosec x)/(cos 2x+ 2cos^2 x))dx = ∫ (( cosec x dx)/(cos^2 x−sin^2 x+2cos^2 x)) = ∫ (( cosec x dx)/(3cos^2 x−sin^2 x)) = ∫ ((cosec x dx)/(3cos^2 x−1+cos^2 x)) = ∫ (1/(sinx ( 4cos^2 x−1 )))dx = ∫ (1/(sin x (4cos^2 x−1)))dx = ∫ (( sin x dx)/(sin^2 x(4cos^2 x−1))) = ∫ (( sin xdx)/((1−cos^2 x)(4cos^2 x−1))) Put , cos x=t sin xdx= −dt ∫ (( −dt)/((1−t^2 )(4t^2 −1)))= ∫ ((−dt)/(−(t^2 −1)(4t^2 −1))) ∫ (( 1)/((t^2 −1)(4t^2 −1)))dt According to partial fraction, −(4/3)∫ (1/(t^2 −1))dt+ (1/3)∫ (1/(4t^2 −1))dt = (1/3)∫(( 1)/((2t)^2 −(1)^2 ))dt−(4/3)∫ (1/((t)^2 −(1)^2 ))dt =(1/3).(1/(2.1)).(1/2)log ((2t−1)/(2t+1)) −(4/3)log ((t−1)/(t+1)) + C = (1/(12)) log ((2cos x−1)/(2cos x+1)) − (4/3) log ((cos x−1)/(cos x+1)) +C$
	$\int \frac{cosec x}{\cos 2 x + {2 \cos}^{2} x} dx$ $= \int \frac{cosec x dx}{\cos^{2} x - \sin^{2} x + {2 \cos}^{2} x}$ $= \int \frac{cosec x dx}{{3 \cos}^{2} x - \sin^{2} x}$ $= \int \frac{cosec x dx}{{3 \cos}^{2} x - 1 + \cos^{2} x}$ $= \int \frac{1}{sinx ({4 \cos}^{2} x - 1)} dx$ $= \int \frac{1}{\sin x ({4 \cos}^{2} x - 1)} dx$ $= \int \frac{\sin x dx}{\sin^{2} x ({4 \cos}^{2} x - 1)}$ $= \int \frac{\sin xdx}{(1 - \cos^{2} x) ({4 \cos}^{2} x - 1)}$ $Put, \cos x = t$ $\sin xdx = - dt$ $\int \frac{- dt}{(1 - t^{2}) ({4 t}^{2} - 1)} = \int \frac{- dt}{- (t^{2} - 1) ({4 t}^{2} - 1)}$ $\int \frac{1}{(t^{2} - 1) ({4 t}^{2} - 1)} dt$ $According to partial fraction,$ $- \frac{4}{3} \int \frac{1}{t^{2} - 1} dt + \frac{1}{3} \int \frac{1}{{4 t}^{2} - 1} dt$ $= \frac{1}{3} \int \frac{1}{{(2 t)}^{2} - {(1)}^{2}} dt - \frac{4}{3} \int \frac{1}{{(t)}^{2} - {(1)}^{2}} dt$ $= \frac{1}{3} . \frac{1}{2.1} . \frac{1}{2} \log \frac{2 t - 1}{2 t + 1} - \frac{4}{3} \log \frac{t - 1}{t + 1} + C$ $= \frac{1}{12} \log \frac{2 \cos x - 1}{2 \cos x + 1} - \frac{4}{3} \log \frac{\cos x - 1}{\cos x + 1} + C$
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