What is the integral of (x^5-4x^3+8x^2)/(x^3+8) ?

The integral of (x^5-4x^3+8x^2)/(x^3+8) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^5-4x^3+8x^2)/(x^3+8)

\int \frac{x ^{5} - 4 x ^{3} + 8 x ^{2}}{x ^{3} + 8} d x

\frac{x ^{3}}{3} - 4 x + \frac{8}{3} ln ∣ x + 2 ∣ + \frac{8}{3} arctan (\frac{1}{3} (x - 1)) - \frac{4}{3} ln \frac{1}{3} (x - 1)^{2} + 1 + C

Solution steps

\int \frac{x ^{5} - 4 x ^{3} + 8 x ^{2}}{x ^{3} + 8} d x

Expand

\frac{x ^{5} - 4 x ^{3} + 8 x ^{2}}{x ^{3} + 8} : \frac{x ^{5}}{x ^{3} + 8} - \frac{4 x ^{3}}{x ^{3} + 8} + \frac{8 x ^{2}}{x ^{3} + 8}

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 ^{5}}{x ^{3} + 8} d x - \int \frac{4 x ^{3}}{x ^{3} + 8} d x + \int \frac{8 x ^{2}}{x ^{3} + 8} d x

\int \frac{x ^{5}}{x ^{3} + 8} d x = \frac{x ^{3}}{3} - \frac{8}{3} ln x^{3} + 8

\int \frac{4 x ^{3}}{x ^{3} + 8} d x = 4 (- 8 (\frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1)))) + x)

\int \frac{8 x ^{2}}{x ^{3} + 8} d x = \frac{8}{3} ln x^{3} + 8

= \frac{x ^{3}}{3} - \frac{8}{3} ln x^{3} + 8 - 4 (- 8 (\frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1)))) + x) + \frac{8}{3} ln x^{3} + 8

Simplify

\frac{x ^{3}}{3} - \frac{8}{3} ln x^{3} + 8 - 4 (- 8 (\frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1)))) + x) + \frac{8}{3} ln x^{3} + 8 : \frac{x ^{3}}{3} - 4 x + \frac{8}{3} ln ∣ x + 2 ∣ + \frac{8}{3} arctan (\frac{1}{3} (x - 1)) - \frac{4}{3} ln \frac{1}{3} (x - 1)^{2} + 1

= \frac{x ^{3}}{3} - 4 x + \frac{8}{3} ln ∣ x + 2 ∣ + \frac{8}{3} arctan (\frac{1}{3} (x - 1)) - \frac{4}{3} ln \frac{1}{3} (x - 1)^{2} + 1

Add a constant to the solution

= \frac{x ^{3}}{3} - 4 x + \frac{8}{3} ln ∣ x + 2 ∣ + \frac{8}{3} arctan (\frac{1}{3} (x - 1)) - \frac{4}{3} ln \frac{1}{3} (x - 1)^{2} + 1 + C

\int \frac{1}{x x} d x

\int \frac{cos ( x )}{sin ^{3} ( x ) - sin ( x )} d x

\int (sin (x) + 2) d x

\int 6 x (3 x^{2} + 5)^{6} d x

\int \frac{e ^{- t}}{t ^{2}} d t

What is the integral of (x^5-4x^3+8x^2)/(x^3+8) ?
The integral of (x^5-4x^3+8x^2)/(x^3+8) is