What is the integral from 1 to e^2 of (ln^7(x^2))/x ?

The integral from 1 to e^2 of (ln^7(x^2))/x is 4096

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integral from 1 to e^2 of (ln^7(x^2))/x

\int_{1}^{e^{2}} \frac{ln ^{7} ( x ^{2} )}{x} d x

4096

Solution steps

\int_{1}^{e^{2}} \frac{ln ^{7} ( x ^{2} )}{x} d x

Simplify

\frac{ln ^{7} ( x ^{2} )}{x} : \frac{128 ln ^{7} ( x )}{x}

= \int_{1}^{e^{2}} \frac{128 ln ^{7} ( x )}{x} d x

Take the constant out:

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

= 128 \cdot \int_{1}^{e^{2}} \frac{ln ^{7} ( x )}{x} d x

Apply u-substitution

: \int_{0}^{2} u^{7} d u

= 128 \cdot \int_{0}^{2} u^{7} d u

Apply the Power Rule

: [\frac{u ^{8}}{8}]_{0}^{2}

= 128 [\frac{u ^{8}}{8}]_{0}^{2}

Compute the boundaries:

32

= 128 \cdot 32

Simplify

= 4096

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What is the integral from 1 to e^2 of (ln^7(x^2))/x ?
The integral from 1 to e^2 of (ln^7(x^2))/x is 4096