Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$$\int \frac{x^3-x-1}{(x+1)^2}\,\mathrm{d}x$$

$$\rightarrow \int \left(x+ \frac{2}{x+1} - \frac{1}{\left(x+1\right)^2} -2 \right)\, \mathrm{d}x$$

is there another method other than long division to solve this integral?

share|improve this question
    
The $-x$ and the $-1$ of the numerator are handled by $u$-subsitution and a tangent substitution. Probably the other can be similarly dispatched. Of course, the resulting trig. integrals may well be worse than the solution route you indicate in your post. –  James S. Cook Apr 8 '13 at 0:54
    
What happened to the $x^2$ in the denominator? As written, the numerator has lesser degree than the denominator so division isn't likely to yield anything. –  Mike Apr 8 '13 at 1:32
    
@Mike, thanks, fixed the typo. –  yiyi Apr 8 '13 at 4:42

1 Answer 1

up vote 4 down vote accepted

You can do a trig substitution: let $x=\tan{\theta}$, $dx = \sec^2{\theta}\,d\theta$. You get

$$\int d\theta \: [\tan{\theta}(1-2 \cos^2{\theta}) - \cos^2{\theta}]$$

which comes out to

$$-\log{\cos{\theta}} + \frac{1}{2} \cos{2 \theta} - \frac{\theta}{2} - \frac{1}{4} \sin{2 \theta} + C$$

where $C$ is an integration constant. You then substitute back, details of which I leave to the reader.

share|improve this answer
    
Why did you write the equation behind the $\mathrm{d}\theta$? –  yiyi Apr 8 '13 at 1:15
1  
It's the way I got used to writing integrals. It comes from my background in Optics, where we have multiple integrals commonly and it is useful to write integrals like that to keep track of which integral goes with which variable. –  Ron Gordon Apr 8 '13 at 1:16

Your Answer

 
discard

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.