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Popular >

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

Solution

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

Solution

\frac{1}{3} ln x^{3} + 1 + 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))) + C

Solution steps

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

Expand

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

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

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

\int \frac{2}{x ^{3} + 1} d x = 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})))

= \frac{1}{3} ln x^{3} + 1 + 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})))

Add a constant to the solution

= \frac{1}{3} ln x^{3} + 1 + 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))) + C

Popular Examples

integral of 1/(x^2+4x+7)

\int \frac{1}{x ^{2} + 4 x + 7} d x

integral of x/(x^4+a^4)

\int \frac{x}{x ^{4} + a ^{4}} d x

integral of 1/(1+sqrt(1-x))

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

integral of (x^{-1/3})/(1+x^{2/3)}

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

integral of 4e^{2x}cos(yz)+3y^2

\int 4 e^{2 x} cos (yz) + 3 y^{2} d x

Frequently Asked Questions (FAQ)

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