List of top Mathematics Questions on Some Properties of Definite Integrals asked in KEAM

The value of $\int_{0}^{\frac{\pi}{2}}\frac{\cos^{11}x}{\cos^{11}x+\sin^{11}x}dx$ is equal to} \textit{Note: The original exam paper contained a typographical error ($c\hat{D}$ instead of $dx$). It has been corrected here to permit a valid solution.

Given that $\int_{0}^{1}\tan^{-1}(t)\,dt = \frac{\pi}{4} - \frac{1}{2}\log 2$, then $\int_{0}^{1}\tan^{-1}(1-t)\,dt =$

The value of the integral $\int_{0}^{\frac{\pi}{2}} \log(\tan \theta)\, d\theta$ is

\textbf \( \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \) is equal to:}

$\int_{0}^{\pi/2} \log\left( \frac{\cos x}{\sin x} \right)\, dx$ is equal to:

\( \int_{-\pi}^{\pi} \frac{\sin^2 x}{1+7^x}\, dx \) is equal to