How do you integrate $\int \frac{{ln}^{6} x}{x} d x$ $int ln^(6)x/x dx$ ?

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Answer 1 · 2015-06-08T00:41:53

First, notice that you can rewrite this as:

$\int \frac{{ln}^{6} x}{x} d x = \int {(ln x)}^{6} \cdot \frac{1}{x} d x$ $int ln^6x/xdx = int (lnx)^6*1/xdx$

Now you can see that we can do u-substitution.

Let:
$u = ln x$ $u = lnx$
$d u = \frac{1}{x} d x$ $du = 1/xdx$

$\int {(ln x)}^{6} \cdot \frac{1}{x} d x = \int u^{6} d u$ $int (lnx)^6*1/xdx = intu^6du$

$= \frac{u^{7}}{7} + C$ $= u^7/7 + C$

$= \frac{{ln}^{7} x}{7} + C$ $= ln^7x/7 + C$

How do you integrate ∫ln6xxdxint ln^(6)x/x dx?