What is the integral from 1 to e^3 of (ln^5(x^2))/x ?

The integral from 1 to e^3 of (ln^5(x^2))/x is 3888

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integral from 1 to e^3 of (ln^5(x^2))/x

\int_{1}^{e^{3}} \frac{ln ^{5} ( x ^{2} )}{x} d x

3888

Solution steps

\int_{1}^{e^{3}} \frac{ln ^{5} ( x ^{2} )}{x} d x

Apply u-substitution

: \int_{1}^{e^{6}} \frac{ln ^{5} ( u )}{2 u} d u

= \int_{1}^{e^{6}} \frac{ln ^{5} ( u )}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{1}^{e^{6}} \frac{ln ^{5} ( u )}{u} d u

Apply u-substitution

: \int_{0}^{6} v^{5} d v

= \frac{1}{2} \cdot \int_{0}^{6} v^{5} d v

Apply the Power Rule

: [\frac{v ^{6}}{6}]_{0}^{6}

= \frac{1}{2} [\frac{v ^{6}}{6}]_{0}^{6}

Compute the boundaries:

7776

= \frac{1}{2} \cdot 7776

Simplify

= 3888

35 cos (x)

\frac{d}{d t} (\frac{1 - t ^{2}}{1 + t ^{2}})

x \to - 1 lim (\frac{x ^{2} - 2}{1 - x})

y^{^{'}} = 4 y^{2}

\frac{\partial}{\partial y} (600 - 4 x + 10 x^{2} y)

What is the integral from 1 to e^3 of (ln^5(x^2))/x ?
The integral from 1 to e^3 of (ln^5(x^2))/x is 3888