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Chapter 4: Problem 11

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

Short Answer

Expert verified
The indefinite integral \(\int \frac{(\ln x)^{2}}{x} dx = \frac{1}{3}(\ln x)^3 + C\).

Step by step solution

01

Identify Function for Substitution

Identify the inner function that makes the integral complex. Here, \(\ln x\) is such a function. We make a substitution such that \(u = \ln x\). This simplifies the integral. Differentiate \(u\) with respect to \(x\) to find \(du\), i.e., \(du = \frac{1}{x} dx\).
02

Replace in Integral

Replace \(\ln x\) and \(dx\) in the integral with \(u\) and \(du\) respectively. This transforms the integral into the much simpler form \(\int u^2 du\).
03

Integrate

Evaluate the integral \(\int u^2 du\). This is a basic power rule integral, and its antiderivative is \(\frac{1}{3}u^3 + C\), where \(C\) is the constant of integration.
04

Substitute Back

Finally, substitute \(u\) back into the integral. Since \(u = \ln x\), the final result of the integral is \(\frac{1}{3}(\ln x)^3 + C\).

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