Question

write the integral in terms of ( u ) and ( d u ). then evaluate.
int \frac{(ln (x))^{9}}{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))^{9}}{x} d x=

Explanation:

Step1: Find $du$

Given $u = \ln(x)$, then $du=\frac{1}{x}dx$.

Step2: Substitute $u$ and $du$ into the integral

The integral $\int\frac{(\ln(x))^{9}}{x}dx$ becomes $\int u^{9}du$.

Step3: Integrate $\int u^{9}du$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$), for $n = 9$, we have $\int u^{9}du=\frac{u^{9 + 1}}{9+1}+C=\frac{u^{10}}{10}+C$.

Step4: Substitute back $u=\ln(x)$

Substituting $u=\ln(x)$ into $\frac{u^{10}}{10}+C$, we get $\frac{(\ln(x))^{10}}{10}+C$.

Answer:

$\frac{(\ln(x))^{10}}{10}+C$