Hello everyone! Today we will walk you through how to take the integral of $\csc(x)$. This is a highly unintuitive integral that cannot be solved by normal means, but $\csc(x)$ is also a very common function, so it’s important to know how to integrate it.
The Integral
First, we multiply the integrand of $\displaystyle\int\csc(x)\,dx$ by $\dfrac{\csc(x)+\cot(x)}{\csc(x)+cot(x)}$:
$$ \displaystyle\int \csc (x)\, dx = \int \dfrac{\csc(x)^2+\csc(x)\cot(x)}{\csc(x)+\cot(x)}dx. $$
This is not something you would probably think of, and it is very difficult to see why this would be useful. However, as we will see later, the derivative of $\csc(x)+\cot(x)$ is quite nice for solving this integral.
Next, we will use $u$-substitution with $u = \csc(x)+\cot(x)$ because $du = (-\csc(x)^2-\csc(x)\cot(x))\,dx$:
$$\int \dfrac{\csc(x)^2+\csc(x)\cot(x)}{\csc(x)+\cot(x)}dx = -\int \dfrac{1}{u}du = -\ln|u| + C = -\ln|\csc(x)+\cot(x)| + C.$$
As we can see, an unintuitive manipulation turned out to be the key to solving this integral. This technique can be used in some other integrals, where you multiply both the numerator and denominator of a fraction by some function to make it easier to solve. Good luck on your future math adventures!
