$\int_0^a f(x) \, dx = $

$ \int_{0}^{\frac{\pi}{2}} \frac{\tan ^{7} x}{\cot ^{7} x+\tan ^{7} x} d x $ is equal to

If $\int_0^{\frac{\pi}{4}} \frac{\sin^2 x}{1+\sin x \cos x} dx = \frac{1}{a} \log_e\left(\frac{a}{3}\right) + \frac{\pi}{b \sqrt{3}}$,where $a, b \in N$,then $a+b$ is equal to ....................

Evaluate the definite integral: $\int_0^\pi \frac{x \cos^2 x}{1+\sin x} dx$

Evaluate the integral: $\int_0^{50 \pi} \sqrt{1-\cos 2x} \, dx$ (in $\sqrt{2}$)

The value of $\int_{-2012}^{2012} (\sin(x^3) + x^5 + 1) dx$ is