MHB Integral of [1/sqrt(8x-x^2)] dx - Daisy Dee's Answer at Yahoo Answers

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The integral of [1/sqrt(8x-x^2)] dx can be evaluated using trigonometric substitution. The domain of the integrand is identified as (0,8). By completing the square, the integral is transformed into a more manageable form. The substitution x-4=4sin(θ) simplifies the integral to I=∫dθ, leading to the result I=sin^(-1)((x-4)/4) + C. This method effectively demonstrates the application of trigonometric identities in solving integrals.
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Here is the question:

Integrate [1/sqrt of (8x-x^2)] dx?

I have posted a link there to this thread so the OP can see my work.
 
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Re: Daisy Dee's question: evaluating a definite integral where a trig. substitution is appropriate

Hello Daisy Dee,

We are given to evaluate:

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

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

Completing the square on the radicand, we obtain:

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

Using the substitution:

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

We may write:

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

For $$-\frac{\pi}{2}<\theta<\frac{\pi}{2}$$ we may write:

$$I=\int\,d\theta=\theta+C$$

Back substituting for $\theta$, we obtain:

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