$\int_2^4 \frac{\log x^2}{\log x^2+\log (36-12x+x^2)} dx$ is equal to

Let $f: [2, 5] \to [2, 5]$ be a bijective function such that $\frac{d}{dx}(f^{-1}(x)) > 0$ for all $x \in [2, 5]$. Then $\int_{2}^{5} (f(x) + f^{-1}(x)) dx$ is

The greater of $\int_0^{\pi /2} \frac{\sin x}{x} \, dx$ and $\frac{\pi}{2}$ is:

Let $m, n, p, q$ be four positive integers. If $\int_0^{2 \pi} \sin^m x \cos^n x \, dx = 4 \int_0^{\pi/2} \sin^m x \cos^n x \, dx$,$\int_0^{2 \pi} \sin^p x \cos^n x \, dx = 0$,$\int_0^{\pi} \sin^p x \cos^q x \, dx = 0$,$a = m + n + p$ and $b = m + n + q$,then:

Let $[\bullet]$ be the greatest integer function. If $\alpha = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) dx$,then $\frac{1}{\pi} \int_{0}^{\alpha\pi} \left( \frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta} \right) d\theta$ is equal to . . . . . . .

The value of $\int_{-100}^{100} \frac{x+x^3+x^5}{1+x^2+x^4+x^6} dx$ is