Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How can I integrate this function ?


share|cite|improve this question
up vote 4 down vote accepted

Divide the $x^2$ by $4x+5$, using ordinary division of polynomials. We get $$\frac{x^2}{4x+5}=\frac{1}{4}x-\frac{5}{16}+\frac{25}{16}\frac{1}{4x+5}.$$ Now the integration should be straightforward.

Alternately, as a second choice, let $u=4x+5$. Then $du=4\,dx$, so $dx=\frac{1}{4}du$. Also, $x=\frac{1}{4}(u-5)$, so $x^2=\frac{1}{16}(u^2-10u+25)$. We end up with $$\int_{u=5}^9 \frac{1}{64}\frac{u^2-10u+25}{u}\,du.$$ The integration is easy, because of the cancellations.

share|cite|improve this answer

Or, alternatively (same idea),

$$\frac{x^2}{4x+5} = \frac{x}{4} + \frac{25}{64x + 80} - \frac{5}{16}$$

which you can probably integrate pretty readily.

share|cite|improve this answer

$$I := \int^1_0 \frac{x^2}{4x+5} \ dx$$

Since the numerator is of a higher degree than the denominator, we need to use long division. Upon long division, we can rewrite our integral as

$$I = \int^1_0 \frac{x}{4} + \frac{25}{16(4x+5)} -\frac{5}{16} \ dx \tag{1}$$

Now simply integrate each piece and let $u = 4x+5, du = 4 \ dx$. Don't forget that

$$\int \frac{du}{u} = \ln |u| + C$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.