What is the integral of (4x+10)/(x^3+x) ?

The integral of (4x+10)/(x^3+x) is 4arctan(x)+10(ln|x|-1/2 ln|x^2+1|)+C

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integral of (4x+10)/(x^3+x)

\int \frac{4 x + 10}{x ^{3} + x} d x

4 arctan (x) + 10 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

Solution steps

\int \frac{4 x + 10}{x ^{3} + x} d x

Expand

\frac{4 x + 10}{x ^{3} + x} : \frac{4 x}{x ^{3} + x} + \frac{10}{x ^{3} + 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{4 x}{x ^{3} + x} d x + \int \frac{10}{x ^{3} + x} d x

\int \frac{4 x}{x ^{3} + x} d x = 4 arctan (x)

\int \frac{10}{x ^{3} + x} d x = 10 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

= 4 arctan (x) + 10 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Add a constant to the solution

= 4 arctan (x) + 10 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

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What is the integral of (4x+10)/(x^3+x) ?
The integral of (4x+10)/(x^3+x) is 4arctan(x)+10(ln|x|-1/2 ln|x^2+1|)+C