Write the integral in terms of $u$ and $d u$. Then evaluate.
\[
\int \frac{(\ln (x))^{6}}{x} d x, \quad u=\ln (x)
\]
(Use symbolic notation and fractions where needed. Use $C$ for the arbitrary constant. Absorb into $C$ as much as possible:)
\[
\int \frac{(\ln (x))^{6}}{x} d x=
\]
Answer
Substitute back in \( u = \ln(x) \) to get the final answer: \(\boxed{\frac{(\ln(x))^7}{7} + C}\)
Step 1 ：Identify the substitution: \( u = \ln(x) \)
Step 2 ：Differentiate both sides with respect to \( x \) to find \( du \): \( du = \frac{1}{x} dx \)
Step 3 ：Rewrite the integral in terms of \( u \) and \( du \): \( \int \frac{(\ln (x))^{6}}{x} dx = \int u^6 du \)
Step 4 ：Integrate \( u^6 \) with respect to \( u \): \( \int u^6 du = \frac{u^7}{7} + C \)
Step 5 ：Substitute back in \( u = \ln(x) \) to get the final answer: \(\boxed{\frac{(\ln(x))^7}{7} + C}\)
