Solving the Indefinite Integral: \int\sqrt{28x-x^2} dx

[talk2glenn]

Homework Statement

Evaluate the indefinite integral.

$\int\sqrt{28x-x^2} dx$

The Attempt at a Solution

$\int\sqrt{196-(x-14)^2} dx$

Completing the square​

$u=x-14$

$du=dx$

$\int\sqrt{196-u^2} du$

u substitution​

$u=14sin\theta$

$du=14cos\theta d\theta$

Trig substitution​

$\int\sqrt{196cos^2\theta} 14cos\theta d\theta$

$\int14cos\theta*14cos\theta d\theta$

$98\int1+cos2\theta d\theta$

$98(\theta+sin\theta*cos\theta) + C$

$98(arcsin(u/14)+(u/14)(\sqrt{196-u^2}/14)$

Solve for Theta​

From here I should be able to just sub in for u and arrive at my answer. Unfortunately, it is close but no correct, and I can't see where I'm going wrong. Any ideas?

 

[Clever-Name]

What's the answer you're getting and what's the answer you're supposed to get. I don't see any errors in what you've written, but perhaps its a simplification error.

 

[micromass]

Hi talk2glenn!

May I know what the exact solution is that you have?µ

I can't see anything wrong with what you did, and wolfram alpha seems to say you're right too! See http://www.wolframalpha.com/input/?i=d%2898*%28arcsin%28%28x-14%29%2F14%29%2B%28%28x-14%29%2F14%29*%28sqrt%28196-%28x-14%29^2%29%29%2F%2814%29%29%29%2Fdx

 

[talk2glenn]

Thanks for the prompt replies :)

So it appears my problem is either data entry or substitution; that's a good sign. Here is my final solution exactly as I've entered it in our software:

98(arcsin((x-14)/14)+(x-14)sqrt(x^2-28x)/2)

Sorry if that's not to pretty; here's the image of the same input:

[PLAIN]http://webwork.asu.edu/webwork2_files/tmp/equations/a6/bd057be8bc635789c21be6e19460f51.png

Comparing it to the Wolfram output, it looks close but not identical. Then again, the computer tends to do some crazy simplification at the end; maybe they're equivalent. See: http://www.wolframalpha.com/input/?i=integrate+sqrt(28x-x^2)

 

[micromass]

Thanks for the prompt replies :)

So it appears my problem is either data entry or substitution; that's a good sign. Here is my final solution exactly as I've entered it in our software:

98(arcsin((x-14)/14)+(x-14)sqrt(x^2-28x)/2)

Doesn't that last term need to be

$$\frac{(x-14)\sqrt{28x-x^2}}{196}$$

So you need to switch the entries in your root around. and I don't really see why you only divide by 2.

Sorry if that's not to pretty; here's the image of the same input:

[PLAIN]http://webwork.asu.edu/webwork2_files/tmp/equations/a6/bd057be8bc635789c21be6e19460f51.png

Comparing it to the Wolfram output, it looks close but not identical. Then again, the computer tends to do some crazy simplification at the end; maybe they're equivalent. See: http://www.wolframalpha.com/input/?i=integrate+sqrt(28x-x^2)

The wolfram output is correct. However, wolfram uses an entirely different algorithm to calculate this. So the answers can look pretty different...

 

[talk2glenn]

You're right; I see my error now. On paper, I distributed the 98*stuff/196 and got stuff/2, but when I entered it I kept the 98 as a factor. Edit: And you're right about the order too- I reversed it. Always rushing these things at the end.

Let me try that and see what happens. Thanks again!