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

解答

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

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

求解步骤

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

提出常数:

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

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

长除

\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

使用积分加法定则:

\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 ∣)

解答补常数

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

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