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

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

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

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

\frac{1}{5} ln x^{2} + 5 + \frac{3}{5} ln ∣ x ∣ + C

Solution steps

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

Expand

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

\int \frac{x ^{2}}{x ^{3} + 5 x} d x = \frac{1}{2} ln x^{2} + 5

\int \frac{3}{x ^{3} + 5 x} d x = 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{10} ln x^{2} + 5)

= \frac{1}{2} ln x^{2} + 5 + 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{10} ln x^{2} + 5)

Simplify

\frac{1}{2} ln x^{2} + 5 + 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{10} ln x^{2} + 5) : \frac{1}{5} ln x^{2} + 5 + \frac{3}{5} ln ∣ x ∣

= \frac{1}{5} ln x^{2} + 5 + \frac{3}{5} ln ∣ x ∣

Add a constant to the solution

= \frac{1}{5} ln x^{2} + 5 + \frac{3}{5} ln ∣ x ∣ + C

\frac{d}{d t} (A e^{- t})

\int e^{π} d x

\frac{d}{d x} (- 16 e^{- x^{4}} x^{3})

\int_{0}^{1} \int_{0}^{y^{2}} x e^{\frac{x}{y ^{2}}} d x d y

x \to 4 lim (\frac{x ^{2} - 3 x - 4}{x - 4})

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