I need to solve the following integral:
$\int_{1/4}^{1/2} \frac{C \cos(\pi x)}{\sin^2(\pi x)} \,dx = 1$
I am not sure about how I can solve this regarding the $C$. Should I solve the integral with the substitution method without calculating the integral of $C$? Or is there something else I should do?
Similar to derivatives, integrals are "linear operators", which means you can factor constants out of them (and distribute them over sums). In other words, $$\begin{align} \int_a^b Cf(x) \,dx &= C \int_a^b f(x) \,dx\\ \int_a^b (f(x) + g(x)) \,dx &= \int_a^b f(x) \,dx + \int_a^b g(x) \,dx \end{align}$$ All this is to say that you can factor $C$ out of the integral then proceed as normal.
In the particular case of the equation you've given, it looks like $C$ plays the role of normalizing the value of the integral to $1$.