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

The integral from 1 to e^5 of (ln^2(x^2))/x is 500/3

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

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

\frac{500}{3}

Decimal Notation

166.66666 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{e^{10}} \frac{ln ^{2} ( u )}{2 u} d u

= \int_{1}^{e^{10}} \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^{10}} \frac{ln ^{2} ( u )}{u} d u

Apply u-substitution

: \int_{0}^{10} v^{2} d v

= \frac{1}{2} \cdot \int_{0}^{10} v^{2} d v

Apply the Power Rule

: [\frac{v ^{3}}{3}]_{0}^{10}

= \frac{1}{2} [\frac{v ^{3}}{3}]_{0}^{10}

Compute the boundaries:

\frac{1000}{3}

= \frac{1}{2} \cdot \frac{1000}{3}

Simplify

= \frac{500}{3}

y = 3 x + sin (5 x)

\frac{d}{d x} ((x^{2} - x)^{4})

\int \frac{x ^{2} + y ^{2} - 5}{( 1 + x ) ^{3}} d x

\frac{d}{d x} (a^{2} + x^{2})

\frac{d v}{d t} = \frac{t}{sin ( v )}

What is the integral from 1 to e^5 of (ln^2(x^2))/x ?
The integral from 1 to e^5 of (ln^2(x^2))/x is 500/3