The question looked fairly awful – STEP questions usually do. It was a request to sketch the function:

$g(x) = \int_{-1}^{1} \frac{1}{\sqrt{1-2xt + x^2}} dt$, for all real $x$.

Have a go, if you’d like! I’ll spoiler it below the line.

### The first insight

The first thing that makes the question less awful than it looks is the realisation that, as far as the integral is concerned, $x$ is a constant. (I suppose it’s a parameter. But it doesn’t change within the integral.)

That means we’re actually trying to integrate $\int_{-1}^{1} \frac{1}{\sqrt{a+bt}} dt$, where $a$ and $b$ are functions of $x$ we don’t much care about – specifically, $a(x) = 1+x^2$ and $b(x)=-2x$.

Writing the integral as $\int_{-1}^{1} (a+bt)^{-1/2} dt$ makes is simple enough to integrate as $\left[ \frac{2}{b} (a+bt)^{1/2} \right]_{-1}^1$, or $\frac{2}{b}\left( \sqrt{a+b} - \sqrt{a-b}\right)$.

Substituting in the values of $a$ and $b$ gives us $-\frac{1}{x} \left(\sqrt{1-2x + x^2} - \sqrt{1+2x+x^2}\right)$

And this is where I fell into…

### The heffalump trap

Aha!, I thought, cleverly: those are both perfect squares! That means, $g(x) = -\frac{1}{x}\left((1-x)-(1+x)\right)$ , which is 2. Job done! Simple sketch, straight line.

Do you see my error? Don’t worry if not, but award yourself a smug pat on the back if so.

$\sqrt{z^2}$ is not $z$. It’s $| z |$. You could happily put 3 into $\sqrt{1-2x+x^2}$ and get 2 out – but $(1-x)$ is -2.

The correct statement of $g(x)$ is $-\frac{1}{x}\left( | 1-x | - | 1+x|\right)$. That’s a much nastier one to sketch.

When $1-x$ and $1+x$ are both positive – that is, for $-1<x<1$, then we do get $g(x)=2$. But only over that domain.

When $x < -1$, $g(x) = -\frac{1}{x}\left( (1-x)+(1+x)\right)$, which simplifies to $g(x) = -\frac{2}{x}$.

Similarly, when $x > 1$, $g(x)$ works out to be $\frac{2}{x}$.

So instead of a straight line, we have a pedestal: outside of the straight line segment from $(-1,2)$ to $(1,2)$, we have two reciprocal curves that drop ever so pleasantly towards the axes.

I thought that was an interesting mistake! Would it have tripped you up?