Question

evaluate the indefinite integral. (use $c$ for the constant of integration.)
int \frac{(ln x)^{28}}{x} dx

Explanation:

Step1: Substitute $u = \ln x$

Let $u=\ln x$, then $du=\frac{1}{x}dx$. The integral $\int\frac{(\ln x)^{28}}{x}dx$ becomes $\int u^{28}du$.

Step2: Integrate $u^{28}$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$), for $n = 28$, we have $\int u^{28}du=\frac{u^{29}}{29}+C$.

Step3: Substitute back $u=\ln x$

Substituting $u = \ln x$ back into the result, we get $\frac{(\ln x)^{29}}{29}+C$.

Answer:

$\frac{(\ln x)^{29}}{29}+C$