# How do you find the definite integral of int (x^2-2)/(x+1) from [0,2]?

Jan 7, 2017

$- \ln 3$

#### Explanation:

Start by dividing the numerator by the denominator using long or synthetic division. Thus:

${\int}_{0}^{2} \frac{{x}^{2} - 2}{x + 1} \mathrm{dx} = {\int}_{0}^{2} x - 1 + - \frac{1}{x + 1} \mathrm{dx}$

We can integrate using the rules $\int \left(\frac{1}{x}\right) \mathrm{dx} = \ln | x | + C$ and $\int \left({x}^{n}\right) \mathrm{dx} = {x}^{n + 1} / \left(n + 1\right) + C$.

$= {\left[\frac{1}{2} {x}^{2} - x - \ln | x + 1 |\right]}_{0}^{2}$

Evaluate using ${\int}_{a}^{b} F \left(x\right) = f \left(b\right) - f \left(a\right)$, where $f ' \left(x\right) = F \left(x\right)$.

$= \frac{1}{2} {\left(2\right)}^{2} - 2 - \ln | 3 | - \left(\frac{1}{2} {\left(0\right)}^{2} - 0 - \ln | 0 + 1 |\right)$

$= \frac{1}{2} \left(4\right) - 2 - \ln | 3 | - 0$

$= - \ln 3$

In celebration of this being the 2000th answer I ever wrote for socratic, I've included a whole bunch of practice exercises for your improvement

Practice exercises

1. Evaluate each definite integral. Round answers to the nearest integer.

a) ${\int}_{1}^{4} \frac{{x}^{3} + 7 x + 14}{x + 2} \mathrm{dx}$

b) ${\int}_{7}^{15} \frac{2 {x}^{4} - 18 {x}^{3} + 2 {x}^{2} - 5 x + 1}{2 x + 1}$

c) ${\int}_{3}^{6} \frac{5 {x}^{5} + 2 {x}^{2} - 8}{2 \left(x + 4\right)}$

Bonus
Hint: Use substitution
b) ${\int}_{e}^{e + 3} \frac{{e}^{2 x} - {e}^{x}}{\sqrt{{e}^{x}}} \mathrm{dx}$

Hopefully this helps!

$1. 33$
$2. 2920$
$3. 2102$
bonus: $3474$ 