/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Find the indefinite integral. ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 44

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

Short Answer

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

Step by step solution

01

Recall the Identity

Knowing the relationships among trigonometric functions is important. Identify that \(\csc ^{2}(x)=1+\cot ^{2}(x)\)
02

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 \(dx = -2 \, du / \csc ^{2}(\frac{x}{2})\).
03

Transform the Integral with the Substitution

Substitute \(u\) and \(dx\) into the original integral: \( \int \csc ^{2}\left(\frac{x}{2}\right) d x = -2 \int u^2 du\)
04

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.
05

Reverse the Substitution

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

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