# Thread: Integration using Partial Fractions

1. ## Integration using Partial Fractions

Hey, I need some help with this integral

$\int\frac{1}{\sqrt{-7+8x-x^2}}$

Thanks

2. Observe that $\int\frac{dx}{\sqrt{-7+8x-x^2}}=\int\frac{dx}{\sqrt{9-(x-4)^2}}=\frac{1}{3}\int\frac{dx}{\sqrt{1-\left(\frac{x-4}{3}\right)^2}}$. So let $\sin t=\frac{x-4}{3}\Rightarrow \cos t = \frac{1}{3}$, then $\frac{1}{3}\int\frac{dx}{\sqrt{1-\left(\frac{x-4}{3}\right)^2}}=\frac{1}{9}\int\frac{\cos t}{\sqrt{1-\sin^2t}}dt=t+K=\frac{1}{9}\arcsin\left(\frac{x-4}{3}\right)+C$.