What is the integral from 1 to e^4 of (ln^2(x^4))/x ?

The integral from 1 to e^4 of (ln^2(x^4))/x is 1024/3

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integral from 1 to e^4 of (ln^2(x^4))/x

\int_{1}^{e^{4}} \frac{ln ^{2} ( x ^{4} )}{x} d x

\frac{1024}{3}

Decimal Notation

341.33333 \dots

Solution steps

\int_{1}^{e^{4}} \frac{ln ^{2} ( x ^{4} )}{x} d x

Simplify

\frac{ln ^{2} ( x ^{4} )}{x} : \frac{16 ln ^{2} ( x )}{x}

= \int_{1}^{e^{4}} \frac{16 ln ^{2} ( x )}{x} d x

Take the constant out:

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

= 16 \cdot \int_{1}^{e^{4}} \frac{ln ^{2} ( x )}{x} d x

Apply u-substitution

: \int_{0}^{4} u^{2} d u

= 16 \cdot \int_{0}^{4} u^{2} d u

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{0}^{4}

= 16 [\frac{u ^{3}}{3}]_{0}^{4}

Compute the boundaries:

\frac{64}{3}

= 16 \cdot \frac{64}{3}

Simplify

= \frac{1024}{3}

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What is the integral from 1 to e^4 of (ln^2(x^4))/x ?
The integral from 1 to e^4 of (ln^2(x^4))/x is 1024/3