# Is there a better way to integrate this

what is the best way to integrate this functio, i integrated in parts

f(x)=x2/2x+3

i split the top of the fraction and then integrated the x2 half using integration in parts, but its just long and lots of place to make mistakes.

I would try to rid of the top bit. Here is a hint $$x^2=\frac{1}{4}((2x-3)(2x+3) + 9)$$.

Good luck

jambaugh
Gold Member
what is the best way to integrate this functio, i integrated in parts

f(x)=x2/2x+3

i split the top of the fraction and then integrated the x2 half using integration in parts, but its just long and lots of place to make mistakes.

I assume you mean:
$$\frac{x^2}{2x+3}$$

Note the degree of the numerator is higher than that of the denominator. (Improper rational function like an improper fraction.)
You should first carryout long division of the polynomials (and check by re-multiplying).
You will then get a polynomial plus a proper fraction. Since the denom. has degree 1 the numerator (remainder) will be constant and you can integrate via simple variable substitution.

Carry out the long division:
$$2x+3)\overline{x^2 + 0x + 0}$$

You should get:
$$P(x)+ \frac{C}{2x+3}$$
where P is a polynomial

You can easily integrate:
$$\int P(x)dx + C/2 \int \frac{du}{u};\quad u = 2x+3$$