Question

evaluate the indefinite integral. (use c for the constant of integration.)
int \frac{(ln x)^{40}}{x} dx

Explanation:

Step1: Set substitution

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

Step2: Rewrite the integral

The integral $\int\frac{(\ln x)^{40}}{x}dx$ becomes $\int u^{40}du$.

Step3: Integrate using power - rule

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$). So, $\int u^{40}du=\frac{u^{41}}{41}+C$.

Step4: Substitute back

Substitute $u=\ln x$ back into the result, we get $\frac{(\ln x)^{41}}{41}+C$.

Answer:

$\frac{(\ln x)^{41}}{41}+C$