What is the integral of (ln(x^{40}))/x ?

The integral of (ln(x^{40}))/x is 20ln^2(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (ln(x^{40}))/x

\int \frac{ln ( x ^{40} )}{x} d x

20 ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{40} )}{x} d x

Apply u-substitution

: \int \frac{ln ( u )}{40 u} d u

= \int \frac{ln ( u )}{40 u} d u

Take the constant out:

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

= \frac{1}{40} \cdot \int \frac{ln ( u )}{u} d u

Apply u-substitution

: \int v d v

= \frac{1}{40} \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= \frac{1}{40} \cdot \frac{v ^{2}}{2}

Substitute back

= \frac{1}{40} \cdot \frac{ln ^{2} ( x ^{40} )}{2}

Simplify

\frac{1}{40} \cdot \frac{ln ^{2} ( x ^{40} )}{2} : 20 ln^{2} (x)

= 20 ln^{2} (x)

Add a constant to the solution

= 20 ln^{2} (x) + C

\int \frac{7 ^{3} x ^{5}}{2} d x

\int e^{2 s i n (2 x)} (2 cos (2 x)) d x

\int 3 x^{2} e^{- 4 x} d x

\int arctan (- x) d x

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

What is the integral of (ln(x^{40}))/x ?
The integral of (ln(x^{40}))/x is 20ln^2(x)+C