Problem 44 - Essential Calculus

Question

Find the indefinite integral. $$ \int \csc ^{2}\left(\frac{x}{2}\right) d x $$

Step-by-Step Solution

Answer

The solution to the integral is \(-\frac{2}{3} \cot ^{3}\left(\frac{x}{2}\right) + C \)

Step 1: Recall the Identity

Knowing the relationships among trigonometric functions is important. Identify that \(\csc ^{2}(x)=1+\cot ^{2}(x)\)

Step 2: Perform a U-Substitution

A u-substitution can simplify the integral. Let \(u=\cot (\frac{x}{2})\). Then, the differential \(du= -\csc ^{2}(\frac{x}{2}) dx / 2\), or \(dx = -2 \, du / \csc ^{2}(\frac{x}{2})\).

Step 3: Transform the Integral with the Substitution

Substitute \(u\) and \(dx\) into the original integral: \( \int \csc ^{2}\left(\frac{x}{2}\right) d x = -2 \int u^2 du\)

Step 4: Calculate New Integral

The new integral can now be solved conventionally: \( -2 \int u^2 du = -2 \times \frac{1}{3} u^3 + C \), where C represents the constant of integration.

Step 5: Reverse the Substitution

Replace \(u\) with the original function to get the solution in terms of x: \( -2 \times \frac{1}{3} u^3 + C = -\frac{2}{3} \cot ^{3}\left(\frac{x}{2}\right) + C \)