# How do I evaluate intsqrt(13+12x-x^2)dx?

Mar 6, 2015

Complete the square, then do a trigonometric substitution.

$\sqrt{13 + 12 x - {x}^{2}} = \sqrt{49 - 36 + 12 x - {x}^{2}} = \sqrt{49 - {\left(x - 6\right)}^{2}}$

So, you want to evaluate $\int \sqrt{49 - {\left(x - 6\right)}^{2}} \mathrm{dx}$.

It's going to get very messy, so I want to let $x - 6 = 7 u$. It is then easy to see that ${\left(x - 6\right)}^{2} = 49 {u}^{2}$ and $\mathrm{dx} = 7 \mathrm{du}$, so the integral becomes $\int 7 \sqrt{49 - 49 {u}^{2}} \mathrm{du} = 7 \int 7 \sqrt{1 - {u}^{2}} \mathrm{du} = 49 \int \sqrt{1 - {u}^{2}} \mathrm{du}$,

Now do a trigonometric substitution. I'll use $u = \sin \theta$ which, of course, makes $\mathrm{du} = \cos \theta d \theta$.

This makes the integral: $49 \int \sqrt{1 - {\sin}^{2} \theta} \cos \theta d \theta$.

This, in turn, becomes $49 \int {\cos}^{2} \theta d \theta$. Now, use the power reduction to re-write ${\cos}^{2} \theta$ as $\frac{1}{2} \left(1 + \cos 2 \theta\right)$.

The problem has become:

Evaluate: $\frac{49}{2} \int \left(1 + \cos 2 \theta\right) d \theta = \frac{49}{2} \left(\theta + \frac{1}{2} \sin 2 \theta\right) + C$

Backfilling: $\theta = {\sin}^{-} 1 u$ and
since $u = \sin \theta$ implies $\cos \theta = \sqrt{1 - {u}^{2}}$

$\frac{1}{2} \sin 2 \theta = \frac{1}{2} \left(2 \sin \theta \cos \theta\right) = u \sqrt{1 - {u}^{2}}$

Thus, the integral evaluates(in terms of $u$) leaving +C for later, to: $\frac{49}{2} \left({\sin}^{-} 1 u + u \sqrt{1 - {u}^{2}}\right) = \frac{1}{2} \left(49 {\sin}^{-} 1 u + 49 u \sqrt{1 - {u}^{2}}\right) = \frac{1}{2} \left(49 {\sin}^{-} 1 \left(\frac{x - 6}{7}\right) + 49 \left(\frac{x - 6}{7}\right) \sqrt{1 - {\left(\frac{x - 6}{7}\right)}^{2}}\right) = \frac{1}{2} \left(49 {\sin}^{-} 1 \left(\frac{x - 6}{7}\right) + 49 \left(\frac{x - 6}{7}\right) \sqrt{1 - {\left(\frac{x - 6}{7}\right)}^{2}}\right) = \frac{1}{2} \left(49 {\sin}^{-} 1 \left(\frac{x - 6}{7}\right) + \left(x - 6\right) \sqrt{49 - {\left(x - 6\right)}^{2}}\right) = \frac{1}{2} \left(49 {\sin}^{-} 1 \left(\frac{x - 6}{7}\right) + \left(x - 6\right) \sqrt{13 + 12 x - {x}^{2}}\right)$

Oh, yeah! Don't forget to put $+ C$.