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Popular >

integral of (x^2+5)/(x^3-8)

Solution

\int \frac{x ^{2} + 5}{x ^{3} - 8} d x

Solution

\frac{1}{3} ln x^{3} - 8 + 5 (\frac{1}{12} ln ∣ x - 2 ∣ - \frac{1}{24} ln x^{2} + 2 x + 4 - \frac{1}{4 3} arctan (\frac{1}{3} (x + 1))) + C

Solution steps

\int \frac{x ^{2} + 5}{x ^{3} - 8} d x

Expand

\frac{x ^{2} + 5}{x ^{3} - 8} : \frac{x ^{2}}{x ^{3} - 8} + \frac{5}{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{5}{x ^{3} - 8} d x

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

\int \frac{5}{x ^{3} - 8} d x = 5 (\frac{1}{12} ln ∣ x - 2 ∣ - \frac{1}{24} ln x^{2} + 2 x + 4 - \frac{1}{4 3} arctan (\frac{1}{3} (x + 1)))

= \frac{1}{3} ln x^{3} - 8 + 5 (\frac{1}{12} ln ∣ x - 2 ∣ - \frac{1}{24} ln x^{2} + 2 x + 4 - \frac{1}{4 3} arctan (\frac{1}{3} (x + 1)))

Add a constant to the solution

= \frac{1}{3} ln x^{3} - 8 + 5 (\frac{1}{12} ln ∣ x - 2 ∣ - \frac{1}{24} ln x^{2} + 2 x + 4 - \frac{1}{4 3} arctan (\frac{1}{3} (x + 1))) + C

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\int (5 x^{2} - \frac{3}{2} x - 4) d x

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\int \frac{3 - x}{x - 4} d x

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\int 3 t sin (t) d t

Frequently Asked Questions (FAQ)

What is the integral of (x^2+5)/(x^3-8) ?
The integral of (x^2+5)/(x^3-8) is