How do you integrate $\frac{{(ln (x))}^{7}}{x} d x$ $((ln(x))^7)/x dx$ ?

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Answer 1 · 2018-03-14T21:24:05

$\frac{1}{8} {(ln (x))}^{8} + C$ $frac(1)(8) (ln(x))^(8) + C$

We have: $\int$ $int$ $\frac{{(ln (x))}^{7}}{x}$ $frac((ln(x))^(7))(x)$ $d x$ $dx$

$= \int$ $= int$ ${(ln (x))}^{7} \cdot \frac{1}{x}$ $(ln(x))^(7) cdot frac(1)(x)$ $d x$ $dx$

Let $u = ln (x) \Rightarrow d u = \frac{1}{x}$ $u = ln(x) Rightarrow du = frac(1)(x)$ $d x$ $dx$ :

$= \int$ $= int$ $u^{7}$ $u^(7)$ $d u$ $du$

$= \frac{u^{7 + 1}}{7 + 1} + C$ $= frac(u^(7 + 1))(7 + 1) + C$

$= \frac{1}{8} u^{8} + C$ $= frac(1)(8) u^(8) + C$

Replace $u$ $u$ with $ln (x)$ $ln(x)$ :

$= \frac{1}{8} {(ln (x))}^{8} + C$ $= frac(1)(8) (ln(x))^(8) + C$

How do you integrate ((ln(x))^7)/x dx(ln(x))7xdx ((ln(x))^7)/x dx?