If $\int \frac{x-\sin x}{1+\cos x} dx = x \tan \left(\frac{x}{2}\right) + p \log \left|\sec \left(\frac{x}{2}\right)\right| + C$,then $p$ is equal to

For $n \ge 2$,if $I_n = \int (\sin x + \cos x)^n dx$,then $nI_n - 2(n-1)I_{n-2} = $

If $I(x) = \int e^{\sin^2 x} (\cos x \sin 2x - \sin x) dx$ and $I(0) = 1$,then $I\left(\frac{\pi}{3}\right)$ is equal to

$\int \frac{d x}{\left(2 a x+x^2\right)^{\frac{3}{2}}} = $

If $\frac{3 \pi}{2} < x < \frac{5 \pi}{2}$ and $\int(\sqrt{1-\sin x}+\sqrt{1+\sin x}) \, dx = f(x) + c$ where $c$ is the constant of integration, then $f\left(\frac{\pi}{3}\right) - f(0) =$

If $\int \frac{\cos x \, dx}{\sin ^{3} x \left(1+\sin ^{6} x\right)^{2 / 3}} = f(x) \left(1+\sin ^{6} x\right)^{1 / \lambda} + c$ where $c$ is a constant of integration,then $\lambda f\left(\frac{\pi}{3}\right)$ is equal to