Write the integral in terms of $u$ and $d u$ . Then evaluate.
$\int \frac{(\ln (x))^{6}}{x} d x, u = \ln (x)$
(Use symbolic notation and fractions where needed. Use $C$ for the arbitrary constant. Absorb into $C$ as much as possible:)
$\int \frac{(\ln (x))^{6}}{x} d x =$

Answer

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Answer

Substitute back in $u = \ln (x)$ to get the final answer: $\frac{(\ln (x))^{7}}{7} + C$

Steps

Step 1 ：Identify the substitution: $u = \ln (x)$

Step 2 ：Differentiate both sides with respect to $x$ to find $d u$ : $d u = \frac{1}{x} d x$

Step 3 ：Rewrite the integral in terms of $u$ and $d u$ : $\int \frac{(\ln (x))^{6}}{x} d x = \int u^{6} d u$

Step 4 ：Integrate $u^{6}$ with respect to $u$ : $\int u^{6} d u = \frac{u^{7}}{7} + C$

Step 5 ：Substitute back in $u = \ln (x)$ to get the final answer: $\frac{(\ln (x))^{7}}{7} + C$