# How do you integrate f(t)=(t^2+2)/(2t-7) using the quotient rule?

Jan 2, 2017

There is no quotient rule for integration (only differentiation). Perform the implied division and integrate the terms of the quotient and the remainder.

#### Explanation:

Use either synthetic or long division to obtain the following:

$\frac{{t}^{2} + 2}{2 t - 7} = \left(\frac{1}{2}\right) t + \frac{7}{4} + \frac{57}{4} \frac{1}{2 t - 7}$

Integrate both sides of the equation:

$\int \frac{{t}^{2} + 2}{2 t - 7} \mathrm{dt} = \left(\frac{1}{2}\right) \int t \mathrm{dt} + \frac{7}{4} \int \mathrm{dt} + \frac{57}{4} \int \frac{1}{2 t - 7} \mathrm{dt}$

$\int \frac{{t}^{2} + 2}{2 t - 7} \mathrm{dt} = \left(\frac{1}{4}\right) {t}^{2} + \frac{7}{4} t + \frac{57}{8} \ln \left(2 t - 7\right) + C$