What is the integral of (x^2+16x-16)/(x^3-16x) ?

The integral of (x^2+16x-16)/(x^3-16x) is 1/2 ln(|x+4||x-4|)-5/2 ln|x+4|+3/2 ln|x-4|+ln|x|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^2+16x-16)/(x^3-16x)

\int \frac{x ^{2} + 16 x - 16}{x ^{3} - 16 x} d x

\frac{1}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) - \frac{5}{2} ln ∣ x + 4 ∣ + \frac{3}{2} ln ∣ x - 4 ∣ + ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{2} + 16 x - 16}{x ^{3} - 16 x} d x

Expand

\frac{x ^{2} + 16 x - 16}{x ^{3} - 16 x} : \frac{x ^{2}}{x ^{3} - 16 x} + \frac{16 x}{x ^{3} - 16 x} - \frac{16}{x ^{3} - 16 x}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{2}}{x ^{3} - 16 x} d x + \int \frac{16 x}{x ^{3} - 16 x} d x - \int \frac{16}{x ^{3} - 16 x} d x

\int \frac{x ^{2}}{x ^{3} - 16 x} d x = \frac{1}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣)

\int \frac{16 x}{x ^{3} - 16 x} d x = 16 (- \frac{1}{8} ln ∣ x + 4 ∣ + \frac{1}{8} ln ∣ x - 4 ∣)

\int \frac{16}{x ^{3} - 16 x} d x = - ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 4 ∣ + \frac{1}{2} ln ∣ x - 4 ∣

= \frac{1}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) + 16 (- \frac{1}{8} ln ∣ x + 4 ∣ + \frac{1}{8} ln ∣ x - 4 ∣) - (- ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 4 ∣ + \frac{1}{2} ln ∣ x - 4 ∣)

Simplify

\frac{1}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) + 16 (- \frac{1}{8} ln ∣ x + 4 ∣ + \frac{1}{8} ln ∣ x - 4 ∣) - (- ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 4 ∣ + \frac{1}{2} ln ∣ x - 4 ∣) : \frac{1}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) - \frac{5}{2} ln ∣ x + 4 ∣ + \frac{3}{2} ln ∣ x - 4 ∣ + ln ∣ x ∣

= \frac{1}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) - \frac{5}{2} ln ∣ x + 4 ∣ + \frac{3}{2} ln ∣ x - 4 ∣ + ln ∣ x ∣

Add a constant to the solution

= \frac{1}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) - \frac{5}{2} ln ∣ x + 4 ∣ + \frac{3}{2} ln ∣ x - 4 ∣ + ln ∣ x ∣ + C

\int \frac{x ^{3}}{8 - x ^{4}} d x

\frac{d}{d x} (arcsinh (x))

\frac{d y}{d x} y^{3} = 7 x^{3} + 2 x

\int (\frac{6}{4 - x ^{2}}) d x

\frac{d}{d y} (e^{x^{2} + y^{2}} sin (x y))

What is the integral of (x^2+16x-16)/(x^3-16x) ?
The integral of (x^2+16x-16)/(x^3-16x) is 1/2 ln(|x+4||x-4|)-5/2 ln|x+4|+3/2 ln|x-4|+ln|x|+C