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

$\int_0^4 \frac{1}{1+\sqrt{x}} \, dx = \dots$

Let $2^{1-a} + 2^{1+a}$, $f(a)$, $3^a + 3^{-a}$ be in $A$.$P$. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\int_{\log_e(\alpha-1)}^{\log_e(\alpha)} \frac{dx}{e^{2x} - e^{-2x}}$ is:

The following integral $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}(2 \operatorname{cosec} x)^{17} d x$ is equal to

Evaluate the definite integral $\int_{0}^{1} \frac{x}{x^{2}+1} d x$.

Evaluate the definite integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x+\cos x}{\sqrt{\sin 2 x}} d x$.