Integration Problem

**looseenz2** · Mar 15th 2010, 06:18 PM

$\int_{0}^{\sqrt3} \frac{5x^2 dx}{x^2 +1}$

Not entirely sure how to do this one.

**mr fantastic** · Mar 15th 2010, 06:20 PM

Originally Posted by looseenz2

$\int_{0}^{\sqrt3} \frac{5x^2 dx}{x^2 +1}$

Not entirely sure how to do this one.

Note that $\frac{5x^2}{x^2 + 1} = 5 \left( \frac{x^2}{x^2 + 1}\right) = 5 \left( \frac{(x^2 + 1) - 1}{x^2 + 1}\right) = 5 - \frac{5}{x^2 + 1}$ .

**looseenz2** · Mar 15th 2010, 06:24 PM

Originally Posted by mr fantastic

Note that $\frac{5x^2}{x^2 + 1} = 5 \left( \frac{x^2}{x^2 + 1}\right) = 5 \left( \frac{(x^2 + 1) - 1}{x^2 + 1}\right) = 5 - \frac{5}{x^2 + 1}$ .

Could you please explain what you did between the last two steps? I'm not entirely sure what you did.

**skeeter** · Mar 15th 2010, 06:46 PM

Originally Posted by looseenz2

Could you please explain what you did between the last two steps? I'm not entirely sure what you did.

the basic rule is ...

when the numerator has a degree $\ge$ the degree of the denominator, divide the fraction.

this can be done with long division (as shown below), or the "shortcut" shown by mr. F

Code:

........1
      ----------
x^2+1 | x^2 
........x^2 + 1
.....  -----------
........... - 1

the quotient is $1 + \frac{-1}{x^2+1} = 1 - \frac{1}{x^2+1}<br />$

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