What is the integral of (x^2+5)/(x^3+4x) ?

The integral of (x^2+5)/(x^3+4x) is -1/8 ln|x^2+4|+5/4 ln|x|+C

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integral of (x^2+5)/(x^3+4x)

\int \frac{x ^{2} + 5}{x ^{3} + 4 x} d x

- \frac{1}{8} ln x^{2} + 4 + \frac{5}{4} ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{2} + 5}{x ^{3} + 4 x} d x

Expand

\frac{x ^{2} + 5}{x ^{3} + 4 x} : \frac{x ^{2}}{x ^{3} + 4 x} + \frac{5}{x ^{3} + 4 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{x ^{2}}{x ^{3} + 4 x} d x + \int \frac{5}{x ^{3} + 4 x} d x

\int \frac{x ^{2}}{x ^{3} + 4 x} d x = \frac{1}{2} ln x^{2} + 4

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

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

Simplify

\frac{1}{2} ln x^{2} + 4 + 5 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4) : - \frac{1}{8} ln x^{2} + 4 + \frac{5}{4} ln ∣ x ∣

= - \frac{1}{8} ln x^{2} + 4 + \frac{5}{4} ln ∣ x ∣

Add a constant to the solution

= - \frac{1}{8} ln x^{2} + 4 + \frac{5}{4} ln ∣ x ∣ + C

(3) (8.6)

\frac{\partial}{\partial y} (\frac{x - y}{y + z})

x \to - 5 lim (\frac{\frac{5}{x} + 1}{x + 5})

\frac{\partial}{\partial x} (x^{\frac{y}{z}})

e^{30 t}

What is the integral of (x^2+5)/(x^3+4x) ?
The integral of (x^2+5)/(x^3+4x) is -1/8 ln|x^2+4|+5/4 ln|x|+C