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Question Number 48239 by olj55336@awsoo.com last updated on 21/Nov/18

$$$$$$$$$$$$$$q.....\int \frac{dx}{\sin x\cos x+2\cos {}^{2}x},pleasesolve$$$$$$

Commented by maxmathsup by imad last updated on 21/Nov/18

$$I=\int \frac{dx}{\frac{1}{2}sin\left(2x\right)+1+cos\left(2x\right)}dx=\int \frac{2dx}{sin\left(2x\right)+2+2cos\left(2x\right)}$$$${=}_{tan\left(x\right)=t}\int \frac{2}{\frac{2t}{1+t^2}+2\frac{1-t^2}{1+t^2}+2}\frac{dt}{1+t^2}$$$$=\int \frac{2dt}{2t+2-2t^2+2+2t^2}=\int \frac{2dt}{2t+4}=\int \frac{dt}{t+2}=ln\mid t+2\mid +C$$$$=ln\mid 2+tan\left(x\right)\mid +C.$$

Answered by tanmay.chaudhury50@gmail.com last updated on 21/Nov/18

$$\int \frac{dx}{{tanxcos}^{2}x+2{cos}^{2}x}$$$$\int \frac{{sec}^{2}xdx}{2+{tan}^{}x}$$$$\int \frac{d(2+tanx)}{2+tanx}$$$$ln(2+tanx)+c$$

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