Methods of Integrals

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$\displaystyle =\pi\left[[\sec x]_{0}^{\frac{\pi}{2}}+[\sec x]_{\frac{\pi}{2}}^{\pi}-[\tan x-x]_{0}^{\pi}\right]$

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$\displaystyle =\frac{1}{2}\int \frac{dx}{\sqrt{\left(\frac{5}{8}\right)^{2}-\left(x-\frac{5}{8}\right)^{2}}}$

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$\displaystyle \textbf{Question 7: } \text{Evaluate } \int_{0}^{\pi} \frac{x\tan x}{\sec x+\tan x}\,dx. \quad$

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$\displaystyle \textbf{Question 9: } \text{Evaluate } \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos x}{1+e^{x}}\,dx.$

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$\displaystyle \therefore I=\int_{0}^{\frac{\pi}{2}}\frac{\cos x}{1+e^{x}}dx+\int_{0}^{\frac{\pi}{2}}\frac{e^{x}\cos x}{1+e^{x}}dx$

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$\displaystyle \text{Let } I=\int_{0}^{\frac{\pi}{2}}\frac{\cos^{2}x}{1+\sin x\cos x}\,dx \quad (i)$

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$\displaystyle \textbf{Question 10: } \text{Evaluate } \int_{0}^{\frac{\pi}{2}} \frac{\cos^{2}x}{1+\sin x\cos x}\,dx. \quad \text{ISC 2018}$

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2. Given the following list of integration techniques:

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Methods of Integrals

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