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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{1}{2 π} e^{- \frac{1}{2} x^{2}} d x

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

\int \frac{1}{x ^{2} \cdot x ^{2} - 16} d x

\int \frac{tan ( 5 x )}{cos ^{2} ( 5 x )} d x

\int e^{3 x} \cdot cos (2 x) d x

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