en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

integral

\frac{2 x + 4}{x ^{3} - 1}

Solution

2 ln ∣ x - 1 ∣ - \frac{2}{3} ln 4 x^{2} + 4 x + 4 - \frac{4}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3})) + C

Solution steps

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

Expand

\frac{2 x + 4}{x ^{3} - 1} : \frac{2 x}{x ^{3} - 1} + \frac{4}{x ^{3} - 1}

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

\int \frac{2 x}{x ^{3} - 1} d x = 2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3})))

\int \frac{4}{x ^{3} - 1} d x = 4 (\frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)))

= 2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3}))) + 4 (\frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)))

Simplify

2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3}))) + 4 (\frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1))) : 2 ln ∣ x - 1 ∣ - \frac{2}{3} ln 4 x^{2} + 4 x + 4 - \frac{4}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3}))

= 2 ln ∣ x - 1 ∣ - \frac{2}{3} ln 4 x^{2} + 4 x + 4 - \frac{4}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3}))

Add a constant to the solution

= 2 ln ∣ x - 1 ∣ - \frac{2}{3} ln 4 x^{2} + 4 x + 4 - \frac{4}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3})) + C

Popular Examples

integral of 3x(1+x)^{-4}integral

3 x (1 + x)^{- 4}

integral of (u^2)/((u^2+1))integral

\frac{u ^{2}}{( u ^{2} + 1 )}

integral of cos^4(x)sin^5(x)integral

cos^{4} (x) sin^{5} (x)

integral of 3^{x+2}integral

3^{x + 2}

integral of 1/((a^2+x^2)^{0.5)}integral

\frac{1}{( a ^{2} + x ^{2} ) ^{0.5}}

Frequently Asked Questions (FAQ)

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