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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Expand

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

\int \frac{3 x}{x ^{3} + 2 x ^{2} + x + 2} d x = 3 (- \frac{2}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (ln x^{2} + 1 + arctan (x)))

\int \frac{1}{x ^{3} + 2 x ^{2} + x + 2} d x = \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (- \frac{1}{2} ln x^{2} + 1 + 2 arctan (x))

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

Simplify

3 (- \frac{2}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (ln x^{2} + 1 + arctan (x))) + \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} (- \frac{1}{2} ln x^{2} + 1 + 2 arctan (x)) : - ln ∣ x + 2 ∣ + \frac{1}{2} ln x^{2} + 1 + arctan (x)

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

Add a constant to the solution

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

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

\int 2 e^{- 2 t} + 4 t + 1 d t

\int - x sin (\frac{x}{2}) cos (\frac{x}{2}) d x

\int \frac{( 1 - r ^{2} ) ^{\frac{5}{2}}}{r ^{8}} d r

\int x - x^{2} + 1 d x

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