What is the integral of (ln(x^4))/(x^3) ?

The integral of (ln(x^4))/(x^3) is -(2ln(x))/(x^2)-1/(x^2)+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^4))/(x^3)

\int \frac{ln ( x ^{4} )}{x ^{3}} d x

- \frac{2 ln ( x )}{x ^{2}} - \frac{1}{x ^{2}} + C

Solution steps

\int \frac{ln ( x ^{4} )}{x ^{3}} d x

Simplify

ln (x^{4}) : 4 ln (x)

= \int \frac{4 ln ( x )}{x ^{3}} d x

Take the constant out:

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

= 4 \cdot \int \frac{ln ( x )}{x ^{3}} d x

Apply Integration By Parts

: - \frac{ln ( x )}{2 x ^{2}} - \int - \frac{1}{2 x ^{3}} d x

= 4 (- \frac{ln ( x )}{2 x ^{2}} - \int - \frac{1}{2 x ^{3}} d x)

\int - \frac{1}{2 x ^{3}} d x = \frac{1}{4 x ^{2}}

= 4 (- \frac{ln ( x )}{2 x ^{2}} - \frac{1}{4 x ^{2}})

Simplify

4 (- \frac{ln ( x )}{2 x ^{2}} - \frac{1}{4 x ^{2}}) : - \frac{2 ln ( x )}{x ^{2}} - \frac{1}{x ^{2}}

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

Add a constant to the solution

= - \frac{2 ln ( x )}{x ^{2}} - \frac{1}{x ^{2}} + C

\int \frac{3}{1 + 9 y ^{2}} d y

\int 500 - \frac{x}{3} d x

\int \frac{- x}{2} d x

\int \frac{11}{a x + d} d x

\int \frac{6 + 3 x}{1 + x ^{2}} d x

What is the integral of (ln(x^4))/(x^3) ?
The integral of (ln(x^4))/(x^3) is -(2ln(x))/(x^2)-1/(x^2)+C