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Question Number 88586 by M±th+et£s last updated on 11/Apr/20

$$\int \frac{\sqrt{cos\left(2x\right)+3}}{cos\left(x\right)}dx$$

Answered by TANMAY PANACEA. last updated on 11/Apr/20

$$\int \frac{cos2x+3}{cosx\sqrt{3+cos2x}}dx$$$$\int \frac{2{cos}^{2}x+2}{cosx\sqrt{2{cos}^{2}x+2}}$$$$\frac{I}{2}=\int \frac{cosx}{\sqrt{2({cos}^{2}x+1)}}dx+\int \frac{secxdx}{\sqrt{2{cos}^{2}x+2}}$$$$=\frac{1}{\sqrt{2}}\int \frac{cosx}{\sqrt{2-{sin}^{2}x}}dx+\frac{1}{\sqrt{2}}\int \frac{{sec}^{2}xdx}{\sqrt{1+{sec}^{2}x}}$$$$\frac{I}{\sqrt{2}}=\int \frac{d\left(sinx\right)}{\sqrt{2-{sin}^{2}x}}+\int \frac{d\left(tanx\right)}{\sqrt{2+{tan}^{2}x}}$$$$I=\sqrt{2}[{sin}^{-1}\left(\frac{sinx}{\sqrt{2}}\right)+ln(tanx+\sqrt{2+{tan}^{2}x})$$$$plscheck$$$$$$

Commented by M±th+et£s last updated on 11/Apr/20

$$correctsolutionthankyousir$$

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