What is the integral of (10x+15)/(x^2+3x) ?

The integral of (10x+15)/(x^2+3x) is 5ln|x|+5ln|x+3|+C

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integral of (10x+15)/(x^2+3x)

\int \frac{10 x + 15}{x ^{2} + 3 x} d x

5 ln ∣ x ∣ + 5 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{10 x + 15}{x ^{2} + 3 x} d x

Take the partial fraction of

\frac{10 x + 15}{x ^{2} + 3 x} : \frac{5}{x} + \frac{5}{x + 3}

= \int \frac{5}{x} + \frac{5}{x + 3} d 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{5}{x} d x + \int \frac{5}{x + 3} d x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

\int \frac{5}{x + 3} d x = 5 ln ∣ x + 3 ∣

= 5 ln ∣ x ∣ + 5 ln ∣ x + 3 ∣

Add a constant to the solution

= 5 ln ∣ x ∣ + 5 ln ∣ x + 3 ∣ + C

\frac{d y}{d x} = \frac{2 - x}{y - 3}

\int \frac{1}{x ( x - 1 ) ( x - 2 )} d x

\frac{\partial}{\partial v} (\frac{v}{u})

f (x) = (2 x)^{4 x}

y^{^{''}} + 16 y = 2 sin (2 t)

What is the integral of (10x+15)/(x^2+3x) ?
The integral of (10x+15)/(x^2+3x) is 5ln|x|+5ln|x+3|+C