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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Long division

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

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

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 q - \int \frac{12}{5 ( q + 60 )} d q)

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

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

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

Add a constant to the solution

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

\int_{0}^{1} 40 x d x

x \to 0 lim (+ (6 - \frac{1}{x ^{3}}))

n = 3 \sum \infty (\frac{8}{11})^{- n}

\int (t^{2} + 4 t) d t

n = 1 \sum \infty \frac{1}{2 ^{n}}

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