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

The integral from 1 to e^2 of (ln^5(x^2))/x is 1024/3

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{1024}{3}

Decimal Notation

341.33333 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{e^{4}} \frac{ln ^{5} ( u )}{2 u} d u

= \int_{1}^{e^{4}} \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^{4}} \frac{ln ^{5} ( u )}{u} d u

Apply u-substitution

: \int_{0}^{4} v^{5} d v

= \frac{1}{2} \cdot \int_{0}^{4} v^{5} d v

Apply the Power Rule

: [\frac{v ^{6}}{6}]_{0}^{4}

= \frac{1}{2} [\frac{v ^{6}}{6}]_{0}^{4}

Compute the boundaries:

\frac{2048}{3}

= \frac{1}{2} \cdot \frac{2048}{3}

Simplify

= \frac{1024}{3}

f (x) = (4 x - 5) (3 x^{3} - x^{2} + 1)

\int \frac{x - 3}{x ^{2} - 1} d x

x^{3}

\int \frac{2 x ^{2} + 1}{x} d x

x \to 1 lim (\frac{x ^{4} - 1}{x ^{2}})

What is the integral from 1 to e^2 of (ln^5(x^2))/x ?
The integral from 1 to e^2 of (ln^5(x^2))/x is 1024/3