$\int_0^{\pi /4} \tan^6 x \sec^2 x \, dx = $

If $\int \limits_0^1 (x^{21}+x^{14}+x^7)(2x^{14}+3x^7+6)^{1/7} dx = \frac{1}{l}(11)^{m/n}$ where $l, m, n \in N$,$m$ and $n$ are coprime,then $l+m+n$ is equal to $...........$.

The value of $\int_1^{e^2} \frac{dx}{x(1 + \ln x)^2}$ is

The integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{d x}{\sin 2 x(\tan ^5 x+\cot ^5 x)}$ is equal to

If $k \int_{0}^{1} x \cdot f(3x) \, dx = \int_{0}^{3} t \cdot f(t) \, dt$,then the value of $k$ is

$\int_0^{\frac{\pi}{4}} \frac{\cos^2 x \sin^2 x}{(\cos^3 x + \sin^3 x)^2} \, dx =$