What is the integral of (5000x)/(25x+1500) ?

The integral of (5000x)/(25x+1500) is 5000(1/25 x-12/5 ln|x+60|)+C

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integral of (5000x)/(25x+1500)

\int \frac{5000 x}{25 x + 1500} d x

5000 (\frac{1}{25} x - \frac{12}{5} ln ∣ x + 60 ∣) + C

Solution steps

\int \frac{5000 x}{25 x + 1500} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5000 \cdot \int \frac{x}{25 x + 1500} d x

Long division

\frac{x}{25 x + 1500} : \frac{1}{25} - \frac{12}{5 ( x + 60 )}

= 5000 \cdot \int \frac{1}{25} - \frac{12}{5 ( x + 60 )} 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

= 5000 (\int \frac{1}{25} d x - \int \frac{12}{5 ( x + 60 )} d x)

\int \frac{1}{25} d x = \frac{1}{25} x

\int \frac{12}{5 ( x + 60 )} d x = \frac{12}{5} ln ∣ x + 60 ∣

= 5000 (\frac{1}{25} x - \frac{12}{5} ln ∣ x + 60 ∣)

Add a constant to the solution

= 5000 (\frac{1}{25} x - \frac{12}{5} ln ∣ x + 60 ∣) + C

x \to 0 lim (\frac{1}{4 x})

\frac{d}{d x} (cos (2 x + 5))

\frac{d}{d x} (\frac{5 x ^{4} - 3 x ^{3}}{x ^{5}})

\int \frac{1}{4} (2 x + 1) d x

\frac{d}{d x} (2 - y - 2 x)

What is the integral of (5000x)/(25x+1500) ?
The integral of (5000x)/(25x+1500) is 5000(1/25 x-12/5 ln|x+60|)+C