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Chapter 8: Problem 26

Find the indefinite integral. $$ \int \frac{\sin \sqrt{x}}{\sqrt{x}} d x $$

Short Answer

Expert verified
The indefinite integral of \(\frac{\sin \sqrt{x}}{\sqrt{x}}\) dx is \(-2\cos \sqrt{x} + C\).

Step by step solution

01

Substitution

Let \(u = \sqrt{x}\), then \(x = u^2\). The differential \(dx = 2u du\). Substituting these values into the integral yields: \(\int \frac{\sin u }{u} * 2u \, du = 2 \int \sin u \, du\).
02

Evaluate Integral

The integral of \(\sin u\) is \(-\cos u\), so our expression becomes: \(2 * -\cos u = -2\cos u\).
03

Back-Substitution

Replace \(u\) with \(\sqrt{x}\) to return to the original variable: \(-2\cos \sqrt{x}.\)
04

Adding Integration Constant

Lastly, remember to add the constant of integration, \(C\), to the solution. The final answer becomes: \(-2\cos \sqrt{x} + C\)

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Most popular questions from this chapter

sketch the graph of the function. $$ y=\csc \frac{2 x}{3} $$

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