Which method should I use for integrating (x^2+x)^-1?

[TylerH]

I'm learning integration by parts, and thought this would be a good test of my understanding.

I've separated it into something that seems better.
$$\int \frac{1}{x^2+x}dx = \int \frac{1}{x} \frac{1}{x+1}dx$$

I'm guessing I use integration by parts from here, but which should I make u?

 

[chiro]

Have you thought about completing the square and then using a trigonometric substitution:

ie x^2 + x + 1/4 - 1/4 = (x + 1/2)^2 - (1/2)^2

 

[PhDorBust]

Partial fractions would be a good approach.

$${1\over x(x+1)} = {1\over x} - {1\over x+1}$$

 

[TylerH]

chiro: That was how Wolfram Alpha did it. It was the hyperbolic tangent that scared me away. :)

PhDorBust: That's awesome. Directly to the solution, too. Could you explain how you spotted that?

 

[lurflurf]

^Partial fractions is a standard method that used to be learned in elementary algebra. Now it has been pushed into elementary calculus elementary ordinary differential equations or e, usually in chapters with names like "more integrals yay!" and "Laplace transforms woo!" respectively.

 

[TylerH]

And I thought I hated U substitution...

 

[dextercioby]

Well, I suggest you do the integral through both methods and compare the results. You'll get an expression for the arctanh in terms of the natural logarithm function which is somewhat expected, since the tanh is defined in terms of the exponential in base <e>.