Questions: Find the indefinite integral. ∫ (ln x)^6 / x dx ∫ (ln x)^6 / x dx = □

Solution Steps

To solve the indefinite integral \(\int \frac{(\ln x)^6}{x} dx\), we can use the substitution method. Let \(u = \ln x\). Then, \(du = \frac{1}{x} dx\). This transforms the integral into a simpler form that can be integrated directly.

1. Use the substitution \(u = \ln x\), which implies \(du = \frac{1}{x} dx\).
2. Rewrite the integral in terms of \(u\).
3. Integrate the resulting expression with respect to \(u\).
4. Substitute back \(u = \ln x\) to get the final answer.

We start with the integral

\[ \int \frac{(\ln x)^6}{x} \, dx. \]

We use the substitution \(u = \ln x\), which gives us \(du = \frac{1}{x} \, dx\). This transforms the integral into

\[ \int u^6 \, du. \]

Next, we integrate \(u^6\):

\[ \int u^6 \, du = \frac{u^7}{7} + C, \]

where \(C\) is the constant of integration.

Now, we substitute back \(u = \ln x\) into our result:

\[ \frac{(\ln x)^7}{7} + C. \]

Final Answer