Question

In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{\pi / 4}^{3 \pi / 4} \cot x d x$$

Short answer

The final result after evaluating the definite integral is \(\ln |\sin(3\pi / 4)| - \ln |\sin(\pi / 4)|\). You can further simplify by using the properties of logarithms and values of special angles.

Step-by-step solution

Identify the substitution function

We'll substitute \(u\) for \(x\), so we begin by setting \(u = x\). The differential \(du\) is simply \(dx\).

Change the limits of integration

Since we're changing the variable we're integrating with respect to, the limits of integration also need to change. In this case, as \(u = x\), we substitute these into our limits: \(u(\pi / 4) = \pi / 4\) and \(u(3\pi / 4) = 3\pi / 4\). So our new limits of integration are \(\pi / 4\) and \(3\pi / 4\).

Perform the u-substitution

After the substitution, the integral becomes \(\int_{\pi / 4}^{3 \pi / 4} \cot(u) du\).

Solve the integral

The integral of \(\cot(u)\) is \(\ln |\sin(u)|\). So the result of our indefinite integral is \(\ln |\sin(u)|\), evaluated from \(\pi / 4\) to \(3\pi / 4\).

Apply limits for the definite integral

Finally, substitute back the limits of integration back into the integral to obtain the final result: \(\ln |\sin(3\pi / 4)| - \ln |\sin(\pi / 4)|\).