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This is the problem: $$ \int \frac{x}{\sqrt{1-4x^2}}\; \mathrm dx$$ My class is just being introduced to u-substitution and we had to use it to evaluate the following integral. My current understanding is that when you have an integral in the form:
$\int f(g(x))\cdot g'(x)\; \mathrm dx$, you can replace $u$ with $g(x)$ and evaluate $\int f(u) \ \; \mathrm d u$.

I plugged this into Wolfram Alpha and it said replace $u$ with $1-4x^2$ and $du$ with $-8x\ dx$

My question(s): How can I know just by looking that this integral:
$1.$ Needs to use u-substitution
$2.$ Know what to subsitute for $u$ and $du$

Sidenote: I don't need the solution past the substitution, I can evaluate it from there.

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    Well, since it isn't a well known integral, we guess it has to do with $u$-substitution, or integration by parts. However, there is a $x$ there on the numerator, so we know that the $\mathrm{d}u$ is going to be have to $a x \mathrm{d}x$ for some constant $a$. So we guess that $u$ is going to be some quadratic equation of $x$. And there is a thing that looks hard to evaluate on the nominator, so we do this. – 2017-02-16
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    The inside part $(u)$ should most clearly be $1-4x^2$. With practice, you will be able to find more complicated u substitutions, but until then, look at what is the innermost part and what is $dx$. (btw, there is a typo in the title) – 2017-02-16
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    Ahh sorry about that. I have fixed it now. – 2017-02-16
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    see that replacing 1-4x^2 by u^2 helps more – 2017-02-16

2 Answers 2

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My question(s): How can I know just by looking that this integral: 
1. Needs to use u-substitution 
2. Know what to subsitute for u and du

It's mostly by patterns and practice. Some points to look at are:

  • Does it look like some of the standard trig integrals?
  • Does it look like a fraction that you can use partial fractions on?
  • Does it follow the pattern that you could easily apply integration by parts?
  • Is it simple enough that you can apply just u-substitution? If you take the innermost statement does it derive to replace the remaining statement? (i.e. if you choose $u = x$ in your equation does it get rid of $\frac{dx}{\sqrt(1-4x^2)}$ if not does $u= 1-x^2$ get rid of $xdx$?)
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Replace $1-4x^2 = u^2 \implies 2udu = -8xdx$

$$\int \frac{x}{\sqrt{1-4x^2}}dx = -\frac14\int \frac{u}{u}du = - \frac14 \int du.$$

I think this will be more easy substitution.