en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^6)/((x^3+1))

Solution

integral

\frac{x ^{6}}{( x ^{3} + 1 )}

Solution

\frac{x ^{4}}{4} - x + \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})) + C

Solution steps

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

Long division

\frac{x ^{6}}{x ^{3} + 1} : x^{3} - 1 + \frac{1}{x ^{3} + 1}

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

\int x^{3} d x = \frac{x ^{4}}{4}

\int 1 d x = x

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

= \frac{x ^{4}}{4} - x + \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}))

Add a constant to the solution

= \frac{x ^{4}}{4} - x + \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})) + C

Popular Examples

integral of x-((x^2)/2)integral

x - (\frac{x ^{2}}{2})

integral of x^{-1}1 integral

x^{- 1} 1

integral of ((x+2))/((x^2+x^2))integral

\frac{( x + 2 )}{( x ^{2} + x ^{2} )}

integral of 4x+1 integral

4 x + 1

integral of 1/((1+4y^2))integral

\frac{1}{( 1 + 4 y ^{2} )}

Frequently Asked Questions (FAQ)

What is the integral of (x^6)/((x^3+1)) ?
The integral of (x^6)/((x^3+1)) is