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

The integral of (2x^2-5x+2)/(x^3+x) is 2ln|x|-5arctan(x)+C

Popular >

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

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

2 ln ∣ x ∣ - 5 arctan (x) + C

Solution steps

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

Take the partial fraction of

\frac{2 x ^{2} - 5 x + 2}{x ^{3} + x} : \frac{2}{x} - \frac{5}{x ^{2} + 1}

= \int \frac{2}{x} - \frac{5}{x ^{2} + 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 \frac{2}{x} d x - \int \frac{5}{x ^{2} + 1} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= 2 ln ∣ x ∣ - 5 arctan (x)

Add a constant to the solution

= 2 ln ∣ x ∣ - 5 arctan (x) + C

\frac{d}{d x} (\frac{- 2}{( x - 1 ) ^{3}})

\int (1 - 2 x)^{2021} d x

x \to + 4 + lim (ln (x - 4))

\frac{1}{( 1 - x ) ^{3}}, 0

\int \frac{2}{x + 2 + x} d x

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