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

The integral from 1 to e^7 of (ln^2(x^2))/x is 1372/3

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

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

\frac{1372}{3}

Decimal Notation

457.33333 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{e^{14}} \frac{ln ^{2} ( u )}{2 u} d u

= \int_{1}^{e^{14}} \frac{ln ^{2} ( 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^{14}} \frac{ln ^{2} ( u )}{u} d u

Apply u-substitution

: \int_{0}^{14} v^{2} d v

= \frac{1}{2} \cdot \int_{0}^{14} v^{2} d v

Apply the Power Rule

: [\frac{v ^{3}}{3}]_{0}^{14}

= \frac{1}{2} [\frac{v ^{3}}{3}]_{0}^{14}

Compute the boundaries:

\frac{2744}{3}

= \frac{1}{2} \cdot \frac{2744}{3}

Simplify

= \frac{1372}{3}

n = 1 \sum \infty (\frac{3}{n})^{4}

\int \frac{1}{sin ^{2} ( t )} d t

r = 1 0^{2 t}

\frac{\partial}{\partial x} (sin (π (4 x - 3 y)))

n = 1 \sum \infty \frac{( n )}{ln ( n )}

What is the integral from 1 to e^7 of (ln^2(x^2))/x ?
The integral from 1 to e^7 of (ln^2(x^2))/x is 1372/3