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

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

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

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

5 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{2 x + 5}{x ^{2} + x} : \frac{5}{x} - \frac{3}{x + 1}

= \int \frac{5}{x} - \frac{3}{x + 1} 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{3}{x + 1} d x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

\int \frac{3}{x + 1} d x = 3 ln ∣ x + 1 ∣

= 5 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣

Add a constant to the solution

= 5 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣ + C

\int ((1 - \frac{2}{3} x^{2})^{6}) (x) d x

\frac{\partial}{\partial y} (x^{2} + y^{2} - 1)

\frac{d}{d x} ((x^{2} - 5 x + 1) (2 x + 3))

\frac{\partial}{\partial y} (\frac{( a x + b y )}{c x + d y})

y^{^{''}} = e^{(3 t)}

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