Litorial Exercise

Evaluate the indefinite integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{(ln(x){)}^{11}}{x}dx$

Step $1$

We must decide what to choose for $u.$

If $u=f(x),$ then $du=f^{\text{'}}(x)dx,$ and so it is helpful to look; for some expression in ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{(ln(x){)}^{11}}{x}dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(ln(x]{)}^{11}(\frac{2}{x})dx$ for which the derivabive is also present, though perhaps missing a constant factor.

For example, $ln(x)$ is part of this integral, and the derivative of $ln(x)$ is $,$ which is also present.