Expert: Ahmed Salami - 9/26/2009

Question
I'm in calculus 2 and I am having a little trouble with this problem. It must be evaluated through the use of long division.

Evaluate the integral of (x^(2)+2)/(x+3)

Answer
Hi Deanna,
To divide x²+2 by x+3,
a. Divide x² by x to get a quotient of x
b. Use the quotient x to multiply the divisor x+3 to get x²+3x
c. Substract x²+3x from x²+2 to get -3x+2
Now, x²+2/x+3 = x + (-3x+2)/x+3
And so you have to do the dividing process again for the fractional part
a. Divide -3x by x to get a quotient of -3
b. Multiply x+3 by -3 to get -3x-9
c. Substract -3x-9 from -3x+2 to get 11
And now,
x²+2/x+3 = x + (-3x+2)/x+3
        = x + (-3 + 11/x+3)
        = x - 3 + 11/x+3
        = (x - 3) + (11/x+3)
This is basically how long division is done.
Notice that 11/x+3 is in its simplest form and cant be divided further and so is considered as the remainder, x-3 being the quotient of the division.
Therefore,
∫(x²+2/x+3)dx = ∫(x - 3 + 11/x+3)dx
             = x²/2 - 3x + 11.ln(x+3) + C

An alternative and rather easier way to do the division is to say
x²+2/x+3 = x²-3²+3²+2/x+3
        = (x²-3² + 3²+2)/x+3
        = (x²-3² + 9+2)/x+3
        = (x²-3² + 11)/x+3
        = (x²-3²)/x+3 + 11/x+3
        = (x-3) + (11/x+3)

Same result. Good luck.

Regards