What is the integral of (8x^2+10)/(x^3+5x) ?

The integral of (8x^2+10)/(x^3+5x) is 3ln|x^2+5|+2ln|x|+C

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integral of (8x^2+10)/(x^3+5x)

\int \frac{8 x ^{2} + 10}{x ^{3} + 5 x} d x

3 ln x^{2} + 5 + 2 ln ∣ x ∣ + C

Solution steps

\int \frac{8 x ^{2} + 10}{x ^{3} + 5 x} d x

Expand

\frac{8 x ^{2} + 10}{x ^{3} + 5 x} : \frac{8 x ^{2}}{x ^{3} + 5 x} + \frac{10}{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{8 x ^{2}}{x ^{3} + 5 x} d x + \int \frac{10}{x ^{3} + 5 x} d x

\int \frac{8 x ^{2}}{x ^{3} + 5 x} d x = 4 ln x^{2} + 5

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

= 4 ln x^{2} + 5 + 2 ln ∣ x ∣ - ln x^{2} + 5

Simplify

= 3 ln x^{2} + 5 + 2 ln ∣ x ∣

Add a constant to the solution

= 3 ln x^{2} + 5 + 2 ln ∣ x ∣ + C

x^{3} = \frac{2 y ^{5} + 4}{5 y ^{4} y ^{^{'}}}

\int sin (16 x) cos (7 x) d x

\frac{\partial}{\partial x} (x^{7} y^{6})

\int 5 x - 1 d x

\frac{d y}{d x} = 8 - \frac{y}{50}, y (0) = 50

What is the integral of (8x^2+10)/(x^3+5x) ?
The integral of (8x^2+10)/(x^3+5x) is 3ln|x^2+5|+2ln|x|+C