Problem 11

Question

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

Step-by-Step Solution

Verified

Answer

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

1Step 1: 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., $du = \frac{1}{x} dx$.

2Step 2: 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 $\int u^2 du$.

3Step 3: 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.

4Step 4: Substitute Back

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

Problem 11

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