Evaluation of a general trigonometric integral How can I evaluate the integral
$$\int\sin^k(x)\ dx$$
in which I don't know if $k$ is an even or an odd number?

The usual form of the resulting integral depends, for one, on the parity of $k$ (i.e., whether it is even or odd).
If you really want a general formula, you could expand using the complex representation of $\sin$, namely,
$$\sin z = \frac{\exp(iz) - \exp(-iz)}{2i},$$
expand $\sin^k z$ using the binomial theorem, integral term-by-term (which is easy, since each term is an exponential function), then rewrite the resulting complex expression as a sum of sine and cosine functions.
The above technique may or may not be useful. In general, it is better to do the following: If $k$ is even, use the double angle identity to write the integral in terms of $\cos^{k / 2}$, and handle any terms $\cos^{\ell}$ with even $\ell$ that occur with the same method. If $k$ is odd, say, $k = 2 m + 1$, we can write $$\int \sin^k x \,dx = \int (1 - \cos^2 x)^m \sin x \,dx,$$ after which the substitution $u = \cos x$ transforms the integral into one with a polynomial integrand. Any terms $\cos^{\ell} x$ with odd $\ell$ that occur can be handled analogously.
