find the indefinite integral. ∫(ln x)^6 / x dx

Explanation:

Step1: Set substitution

Let $u = \ln x$, then $du=\frac{1}{x}dx$.

Step2: Rewrite integral

The integral $\int\frac{(\ln x)^6}{x}dx$ becomes $\int u^{6}du$.

Step3: Apply power - rule for integration

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). So, $\int u^{6}du=\frac{u^{7}}{7}+C$.

Step4: Substitute back

Substitute $u=\ln x$ back into the result. We get $\frac{(\ln x)^{7}}{7}+C$.

Answer:

$\frac{(\ln x)^{7}}{7}+C$