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I have posted a link there to this thread so the OP can see my work.Integrate [1/sqrt of (8x-x^2)] dx?

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I have posted a link there to this thread so the OP can see my work.Integrate [1/sqrt of (8x-x^2)] dx?

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Hello Daisy Dee,

We are given to evaluate:

\(\displaystyle I=\int\frac{1}{\sqrt{8x-x^2}}\,dx\)

We find that the domain of the integrand is $(0,8)$.

Completing the square on the radicand, we obtain:

\(\displaystyle I=\int\frac{1}{\sqrt{4^2-(x-4)^2}}\,dx\)

Using the substitution:

\(\displaystyle x-4=4\sin(\theta)\,\therefore\,dx=4 \cos(\theta)\,d\theta\)

We may write:

\(\displaystyle I=\int\frac{4\cos(\theta)}{\sqrt{4^2-4^2\sin^2(\theta)}}\,d\theta\)

For \(\displaystyle -\frac{\pi}{2}<\theta<\frac{\pi}{2}\) we may write:

\(\displaystyle I=\int\,d\theta=\theta+C\)

Back substituting for $\theta$, we obtain:

\(\displaystyle I=\sin^{-1}\left(\frac{x-4}{4} \right)+C\)