Question

\(\int \csc x d x \quad(\)Hint\(:\) Multiply the integrand by \(\frac{\csc x+\cot x}{\csc x+\cot x}\) and then use a substitution to integrate the result.)

Short answer

\(-\ln|\csc(x) + \cot(x)|\).

Step-by-step solution

Multiply the integrand by trick fraction

Multiply the integrand by \(\frac{\csc(x) + \cot(x)}{\csc(x) + \cot(x)}\), which is essentially multiplying by 1, so it doesn't change the value of the rational function. However, it allows us to rewrite the function in a more convenient way. After this multiplication, the integral becomes \(\int \frac{\csc^2(x) + \csc(x)\cot(x)}{\csc(x) + \cot(x)} d x\).

Use substitution

Let's define a new variable \(u = \csc(x) + \cot(x)\). Then, the derivative \(du\) can be written as \(-\csc(x)\cot(x) - \csc^2(x) dx\), if we look at the numerator of the resulting integral from step 1, we can see that this is the same as the negative of our \(du\). Thus, the integral can be written in terms of \(u\) as \(-\int \frac{du}{u}\).

Apply logarithmic integration

The integral \(-\int \frac{du}{u}\) is a simple integral that corresponds to the log function. Thus, \(-\int \frac{1}{u} du = -\ln|u|\).

Substitute back original function

Finally, replace \(u\) with \(\csc(x) + \cot(x)\) in the solution to get the final answer. This yields the result \(-\ln|\csc(x) + \cot(x)|\).