What is the integral from 1 to e^2 of (ln^8(x^2))/x ?

The integral from 1 to e^2 of (ln^8(x^2))/x is 131072/9

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^8(x^2))/x

\int_{1}^{e^{2}} \frac{ln ^{8} ( x ^{2} )}{x} d x

\frac{131072}{9}

Decimal Notation

14563.55555 \dots

Solution steps

\int_{1}^{e^{2}} \frac{ln ^{8} ( x ^{2} )}{x} d x

Simplify

\frac{ln ^{8} ( x ^{2} )}{x} : \frac{256 ln ^{8} ( x )}{x}

= \int_{1}^{e^{2}} \frac{256 ln ^{8} ( x )}{x} d x

Take the constant out:

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

= 256 \cdot \int_{1}^{e^{2}} \frac{ln ^{8} ( x )}{x} d x

Apply u-substitution

: \int_{0}^{2} u^{8} d u

= 256 \cdot \int_{0}^{2} u^{8} d u

Apply the Power Rule

: [\frac{u ^{9}}{9}]_{0}^{2}

= 256 [\frac{u ^{9}}{9}]_{0}^{2}

Compute the boundaries:

\frac{512}{9}

= 256 \cdot \frac{512}{9}

Simplify

= \frac{131072}{9}

5 e^{x}

\int 2 x (- e^{- x}) d x

y = 3 x^{2} + 3, (2, 15)

\frac{\partial}{\partial x} (e^{k x - m t})

\frac{d}{d x} (x^{2} arctan (8 x))

What is the integral from 1 to e^2 of (ln^8(x^2))/x ?
The integral from 1 to e^2 of (ln^8(x^2))/x is 131072/9