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February 25

Integrating with the natural logarithm

In my textbook this example is given.
$\int \frac{1}{4x-1}dx=\frac{1}{4}\int (\frac{1}{4x-1})4dx=\frac{1}{4}\int \frac{1}{u}du=\frac{1}{4}\ln (\left|u\right|)+C=\frac{1}{4}\ln (|4x-1|)+C$
Where does the 4 in the denominator of the fraction to the left of the integrand come from? If it had been $\int \frac{1}{4x^2-1}dx$ would the same denominator be 8x on account of differentiation or something? --Melab±1 ☎ 21:33, 25 February 2012 (UTC)

The just multiplied and divided by 4, then used the linearity property of $\int$. The answer to your second question is "no". Widener (talk) 21:42, 25 February 2012 (UTC)

But, I don't understand why I multiply and divide by 4. I don't know where the 4 comes from. If the denominator of the fraction inside the integral was 5x^2+7x-1, what would I multiply and divide by? --Melab±1 ☎ 22:00, 25 February 2012 (UTC)

In rewriting the integrand in the form $\int \frac{1}{u}du$, your $du=4dx$, so $\frac{1}{4}du=dx$. --Kinu ^t/_c 23:24, 25 February 2012 (UTC)

... and to follow up on the second part of your question, the use of the log rule only works because the derivative of $u=4x-1$ is expressible as a multiple of the $dx$ present in the original integrand. In the second example, unless the integrand contains a multiple of $(10x+7)dx$ (i.e., the derivative of $u=5x^2+7x-1$), this method wouldn't work. Chances are this isn't something that would be given in the section of the textbook you're looking at right now, of course. --Kinu ^t/_c 23:30, 25 February 2012 (UTC)

So if $du=2xdx$ then I would multiply by $\frac{1}{2x}$? --Melab±1 ☎ 00:41, 26 February 2012 (UTC)

No, because $\int \frac{f(x^2)}{2x}2xdx\ne \frac{1}{2x}\int f(x^2)2xdx$. To integrate $\frac{1}{4x^2-1}$, notice that $\frac{1}{4x^2-1}=\frac{1}{4x-2}-\frac{1}{4x+2}$ Widener (talk) 01:45, 26 February 2012 (UTC)

Still does not answer the question of how I get the denominator. --Melab±1 ☎ 01:47, 26 February 2012 (UTC)

Make the substitution $u=4x-1$. It then follows that $du=4dx$. Therefore, we need $4dx$ in the integrand. We achieve this by multiplying and dividing the integrand by 4. That's where the 4 comes from. Widener (talk) 01:55, 26 February 2012 (UTC)

To avoid the appearance of the 4 without explanation, an alternative derivation is

$\int \frac{1}{4x-1}dx=\int \frac{1}{u}dx=\int \frac{1}{u}\frac{du}{4}=\frac{1}{4}\int \frac{1}{u}du\dots$

where we use the fact that $du=4dx$ to go from the second to the third expression. Gandalf61 (talk) 09:01, 26 February 2012 (UTC)

I am still confused, so I'll pose it more generally.

$\int \frac{1}{4x-1}dx=\frac{1}{a}\int (\frac{1}{4x-1})4dx=\frac{1}{a}\int (\frac{1}{u})du$

How do I get $a$? The problem I am working on is $\int \frac{x}{x^2+1}dx$. $u=x^2+1$ and $du=2xdx$ so I have:

$\int \frac{x}{x^2+1}dx=\frac{1}{1}\int \frac{(x}{x^2+1})2xdx=\int \frac{x}{u}du$

Now the book emphasizes $\int \frac{u^{\prime }}{u}dx=ln(\left|u\right|)+C$, so I how do I use it in the above. --Melab±1 ☎ 23:47, 26 February 2012 (UTC)

The equality $\int \frac{1}{4x-1}dx=\frac{1}{a}\int (\frac{1}{4x-1})4dx$ obviously holds only if $a=4$.
How did you figure out $\int \frac{x}{x^2+1}dx=\frac{1}{1}\int \frac{(x}{x^2+1})2xdx$? That's obviously false. Widener (talk) 01:47, 27 February 2012 (UTC)

$\int \frac{x}{x^2+1}dx=\frac{1}{2}\int \frac{2x}{x^2+1}dx$ Notice that in the integrand the numerator is now the derivative of the denominator and you can use the theorem emphasized in your book. Widener (talk) 01:52, 27 February 2012 (UTC)

I think I get it now. It has to be the numerator in the integrand must be manipulated to make it the derivative of its denominator. --Melab±1 ☎ 22:42, 27 February 2012 (UTC)

You need to be careful to distinguish between which transformations are valid (because the two sides of the equality are indeed equal), and which of the valid transformations will make progress in solving a problem.

You can multiply and divide a quantity by the same number and it will not change. So

$3=\frac{3}{7}\cdot 7$ and $3x^2-2x=\frac{3x^2-2x}{5}\cdot 5$ and $\int \frac{1}{4x-1}dx=\int \frac{1}{4}\frac{1}{4x-1}\cdot 4dx$.

It is also true in general that for a constant number a, $\int af(x)dx=a\int f(x)dx$, and so $\int \frac{1}{4}\frac{1}{4x-1}\cdot 4dx=\frac{1}{4}\int \frac{1}{4x-1}\cdot 4dx$.

It is just as true to say that $\int \frac{1}{4x-1}dx=\frac{1}{3}\int \frac{1}{4x-1}\cdot 3dx$, the difference is that the latter will not help you solve the problem. The former will, because if you substitute $u=4x-1$ (the denominator), you'll have $4dx=du$.

So for similar problems with a linear denominator, you'll want to multiply and divide by the coefficient of x. But you cannot expect this to work for any kind of denominator - integration is a hard problem and there's no general solution, you need to apply transformations that are both true and bring you closer to your goal. In your $\int \frac{x}{x^2+1}dx$ example, both $\int \frac{x}{x^2+1}dx=\frac{1}{1}\int \frac{x}{x^2+1}2xdx$ and $\int \frac{x}{x^2+1}dx=\frac{1}{2x}\int \frac{x}{x^2+1}2xdx$ are of course completely false. You can use for example $\int \frac{x}{x^2+1}dx=\frac{1}{2}\int \frac{1}{x^2+1}2xdx=\frac{1}{2}\int \frac{1}{u}du$, or you can do a partial fraction decomposition as suggested by Widener (which in this case would involve complex numbers). -- Meni Rosenfeld (talk) 06:23, 27 February 2012 (UTC)