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Popular Calculus >

integral of (x^2+1)/(x^3+8)

Solution

\int \frac{x ^{2} + 1}{x ^{3} + 8} d x

Solution

\frac{1}{3} ln x^{3} + 8 + \frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1))) + C

Solution steps

\int \frac{x ^{2} + 1}{x ^{3} + 8} d x

Expand

\frac{x ^{2} + 1}{x ^{3} + 8} : \frac{x ^{2}}{x ^{3} + 8} + \frac{1}{x ^{3} + 8}

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} + 8} d x + \int \frac{1}{x ^{3} + 8} d x

\int \frac{x ^{2}}{x ^{3} + 8} d x = \frac{1}{3} ln x^{3} + 8

\int \frac{1}{x ^{3} + 8} d x = \frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1)))

= \frac{1}{3} ln x^{3} + 8 + \frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1)))

Add a constant to the solution

= \frac{1}{3} ln x^{3} + 8 + \frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1))) + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of (x^2+1)/(x^3+8) ?
The integral of (x^2+1)/(x^3+8) is