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

The integral of ((x^3+2x^2+5x+4)/(x+2)) is -2x^2+x+1/3 (x+2)^3-6ln|x+2|+10+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

\int (\frac{x ^{3} + 2 x ^{2} + 5 x + 4}{x + 2}) d x

- 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} - 6 ln ∣ x + 2 ∣ + 10 + C

Solution steps

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

Expand

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

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

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

\int \frac{2 x ^{2}}{x + 2} d x = x^{2} - 4 x + 8 ln ∣ x + 2 ∣ - 12

\int \frac{5 x}{x + 2} d x = 5 (x + 2 - 2 ln ∣ x + 2 ∣)

\int \frac{4}{x + 2} d x = 4 ln ∣ x + 2 ∣

= - 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12 + x^{2} - 4 x + 8 ln ∣ x + 2 ∣ - 12 + 5 (x + 2 - 2 ln ∣ x + 2 ∣) + 4 ln ∣ x + 2 ∣

Simplify

- 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12 + x^{2} - 4 x + 8 ln ∣ x + 2 ∣ - 12 + 5 (x + 2 - 2 ln ∣ x + 2 ∣) + 4 ln ∣ x + 2 ∣ : - 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} - 6 ln ∣ x + 2 ∣ + 10

= - 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} - 6 ln ∣ x + 2 ∣ + 10

Add a constant to the solution

= - 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} - 6 ln ∣ x + 2 ∣ + 10 + C

\int \frac{1}{( cos ( x ) + sin ( x ) )} d x

\int x (x^{2} - 3)^{\frac{7}{2}} d x

\int 3 sin (x) + 5 cos (x) d x

\int \frac{( 1 + 2 x ^{2} )}{x} d x

\int cot (5 x - 7) d x

What is the integral of ((x^3+2x^2+5x+4)/(x+2)) ?
The integral of ((x^3+2x^2+5x+4)/(x+2)) is -2x^2+x+1/3 (x+2)^3-6ln|x+2|+10+C