How do I pick the right "u" for u substitution and when do I know I need to use it?

Question

This is the problem I'm working with:

$$\int_{-\infty}^{\infty} \frac{4}{16+x^2}dx$$

So when integrating $4/(16+x^2)$ I know it has to be $\arctan$ something because of the $1/(1+x^2)$ format but how do I exactly know what it is? Like this one guy solved it where he did this:

Let $u=x/4 \Rightarrow 4du=dx$ so the integrand becomes $16/(16+16u^2) = 1/(1+u^2)$ and then it fits right in the $\arctan$ format.

My questions are:

1. What to pick $u$ when it isn’t as clear? (e.g. How do I come up with the $x/4$?)
2. Are there any tricks to spot it at first glance?
3. Are there practice problems to help me gain these skills?
4. When do I need to use $u$-sub?

Answer 1

Unfortunately for everyone there's no general way of doing this. You have to "see it". For most of us which haven't been integrating since kindergarten, a way to "see it" is to do plenty of exercises. That's how you just "know" $\mathbf{arctan}$ is involved (edit: OP said so in the original version post).

Note that the $4$ in the numerator isn't really an issue, you can take that "outside the integral", informally speaking. So essentially we would like to transform $16+x^2$ into something of the sort of $1+u^2$. One could try to first get rid of that $16$. Again, multiplying or dividing by constants isn't an issue because if we know how to integrate $f$ we know how to integrate $16 \cdot f$.

Well, we might then just deal with $(16+x^2)/16 = 1+x^2/16$. This is not quite right, but then we "know" (as in, we've exposed ourselves to this a billion times already) that $16 = 4^2$, so this is in fact $1+(x/4)^2$. Thus taking $u = x/4$ will help, we just have to deal with the constants we set aside along the way.

I don't know how enlightening this is - and I suspect not very much - but I don't know if there's really much more to this than "guessing".

Answer 2

$u$-sub is basically anti-chain rule.

If you can do lots of chain rule differentiations in your head, you will be able to spot the analogous $u$-sub when it arises. For example, if you can easily differentiate $(2x^3+5)^4$, then next time you see “integrate $x(6x^2+1)^7$” you’ll see how it goes.

Of course, that was an easy example. However, this does suggest that a strategy to get better at solving these problems is to do both complicated derivatives (including various combinations of trig, log, exponential, inverse etc) and complicated integrals. The first will give you inspiration on how to do the second.

I’m sure if you’ve ever differentiated $\arctan 4x$ or anything similar (e.g. any inverse trig with some $cx$ input), you’ll be able to see the $u$-sub in your question straight away.

Answer 3

Generally the more integrals you do the faster you will recognise a particular style that fits a given method, so for example: $$\int\frac{dx}{x^2+a^2}\tag{1}$$ you would use the substitution $x=a\tan t$ because we know we want an equation of the form $\tan^2u+1\equiv\sec^2 u$ there are also several others. I will leave a list below of some of my general rules that help a lot.

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$$\int f'(x)e^{f(x)}dx=e^{f(x)}+C$$ $$\int f(x)f'(x)dx=\frac12f^2(x)+C$$ $$\int\frac{f'(x)}{f(x)}dx=\ln|f(x)|+C$$ $$\int_0^\infty t^{x-1}e^{-t}dt=\Gamma(x)$$ $$\int_0^1t^{x-1}(1-t)^{y-1}dt=B(x,y)$$ etc. these are not the same form but they all come from the idea of recognising a style of integral and using a general rule

Answer 4

$u$-substitution is appropriate when you see a function and its derivative within the integrand, like in the following examples:

$$\int{\color{blue}{2x}e^{\color{blue}{x^2}}\,dx}$$

$$\int{3\color{blue}{\sin{x}\cos{x}}\,dx}$$

(Since $(\sin{x})'=\cos{x}$ and $(\cos{x})'=\mbox{-}\sin{x}$, you have two choices for this one.)

Some integrals require some manipulation in order to find a good $u$-substitution:

$$\int{\frac{\ln{x}}{x}\,dx} \Rightarrow \int{\color{blue}{\frac{1}{x}\ln{x}}\,dx}$$

As others have mentioned, LOTS of practice is required to determine the best substitution. Since you asked for some more practice, I recommend working on the practice problems here and here.