Evaluate :
\[ I = \int_{-\pi}^{\pi} \frac{2x\,(1+\sin x)}{1+\cos^2 x}\,dx \]
\[ \int_{0}^{\pi/2} \frac{\sin^{2} x}{1+\sin x \cos x}\, dx \]
If \(\displaystyle\int_{0}^{4}\frac{1}{x+\sqrt{x}}\,dx = k\log 3\), then the value of \(k\) is :
If \( \displaystyle\int_{0}^{1} (9x^{2} - 4x - 2k)\,dx = 0 \), then the value of \( k \) is :
The value of \(\int_{-5}^{-1} \frac{1}{x}\,dx\) is equal to :
\[ \int_0^{\frac{\pi}{2}} \frac{\sin^2 x}{1 + \sin 2x}\,dx \]
If \(\displaystyle\int_{0}^{2a}\frac{1}{1+4x^{2}}\,dx = \frac{\pi}{6}\), then the value of a is