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Popular Calculus >

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

Graph

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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