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

The integral of (x+5)/(x^2+10x) is 1/2 ln|x|+1/2 ln|x+10|+C

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

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

\frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 10 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x + 5}{x ^{2} + 10 x} : \frac{1}{2 x} + \frac{1}{2 ( x + 10 )}

= \int \frac{1}{2 x} + \frac{1}{2 ( x + 10 )} 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{1}{2 x} d x + \int \frac{1}{2 ( x + 10 )} d x

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

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

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 10 ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 10 ∣ + C

\int_{0}^{1} π (1 - x^{2}) d x

\frac{- 5}{s ( s ^{2} - 5 s + 4 )}

y = - 3 x - 2

x^{^{''}} + 6 x^{^{'}} + 7 x = 0, x (0) = 0, x^{^{'}} (0) = 1

\int_{2}^{5} (4 x - 2) d x

What is the integral of (x+5)/(x^2+10x) ?
The integral of (x+5)/(x^2+10x) is 1/2 ln|x|+1/2 ln|x+10|+C