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

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

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

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

2 ln ∣ 2 x + 5 ∣ + 2 ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

= 4 ln ∣ 2 x + 5 ∣ + 10 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ 2 x + 5 ∣)

Simplify

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

= 2 ln ∣ 2 x + 5 ∣ + 2 ln ∣ x ∣

Add a constant to the solution

= 2 ln ∣ 2 x + 5 ∣ + 2 ln ∣ x ∣ + C

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