Problem 43 - Essential Calculus

Question

Use a computer algebra system to find or evaluate the integral. $$ \int_{\pi / 4}^{\pi / 2}(\csc x-\sin x) d x $$

Step-by-Step Solution

Answer

The result of the integral from \(\pi/4\) to \(\pi/2\) of \(\csc{x} - \sin{x}\) is \(-\ln{(\sqrt{2}+1)} + 1\).

Step 1: Identify the antiderivatives

First, identify the antiderivatives of the functions involved: The antiderivative of \(\sin{x}\) is \(-\cos{x}\), and the antiderivative of \(\csc{x}\) is \(-\ln|\csc{x} + \cot{x}|\).

Step 2: Evaluate the definite integrals

Next, evaluate the definite integrals for each of these identified antiderivatives individually. For \(-\cos{x}\), evaluate from \(\pi/4\) to \(\pi/2\) and for \(-\ln|\csc{x} + \cot{x}|\), also evaluate from \(\pi/4\) to \(\pi/2\). The results obtained are then subtracted according to the given integral.

Step 3: Compute the integral values and the difference

Now evaluate the values obtained in step 2 at \(\pi/4\) and \(\pi/2\) to get separate results for each of the integrals. Then, subtract the integral of \(\sin{x}\) from the integral of \(\csc{x}\), and obtain the final result.