The value of $\int_0^{1/2} \frac{dx}{\sqrt{1-x^{2n}}}$ is $(n \in N)$

Find the definite integral of $\int_{a}^{b} \frac{1}{x} dx$.

If ${I_n} = \int_0^\infty {{e^{ - x}}{x^{n - 1}}dx,} $ then $\int_0^\infty {{e^{ - \lambda x}}{x^{n - 1}}dx = } $

Evaluate the definite integral: $\int_{0}^{1} \left(1 - \frac{x}{1!} + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \cdots \infty\right) e^{2x} \, dx$.

Evaluate the integral: $\int_0^\pi \left(\cos^2 \left(\frac{3\pi}{8} - \frac{x}{4}\right) - \cos^2 \left(\frac{11\pi}{8} + \frac{x}{4}\right)\right) dx$

For a real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$ and $\{x\} = x - [x]$. Let $n$ be a positive integer. Then,$\int_0^n \cos(2 \pi [x] \{x\}) dx$ is equal to