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I just finished my exam and there was this question. It asks us for which form in the table of integrals to use.

$\int(x-3)\sqrt(6+6x-x^2)$

I did completing the square and got into this form

$\int(x-3)\sqrt(15-(x-3)^2)$

then u-sub with u= x-3

and I got this form

$\int u\sqrt(15-u^2)$

I know I can solve it using substitution again but the question is a MCQ question. It asks us to choose which one to use from the table of integration.

I think the choices were 29,30,11,12. But neither seems to fit my equation?

http://mgh-images.s3.amazonaws.com/9780073532325/1702-6.5-28EI2.png

Zhi J Teoh
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  • my bad. Here they are http://mgh-images.s3.amazonaws.com/9780073532325/1702-6.5-28EI2.png – Zhi J Teoh Nov 21 '14 at 03:25
  • $\LaTeX$ hint: if you enclose the argument of the square root in braces, the square root sign extends to cover. Compare \sqrt(x^2+1) with \sqrt{x^2+1}. They become $\sqrt(x^2+1)$ and $\sqrt{x^2+1}$ – Ross Millikan Nov 21 '14 at 03:31
  • To be honest, it's sort of a stupid question to ask which entry in the table to use, when clearly there are multiple ways to get the antiderivative. – Nishant Nov 21 '14 at 03:40
  • You are correct, none of them fit your integral. That means (unless you have another table) you need to do something more. You are also correct that doing $u^2=w$ will get you to one of the forms in your table. Did MCQ specify that you stop here? There are two questions: the mathematical one is how to evaluate the integral, given the table you have. The test one is how to produce an acceptable answer. It seems you understand the mathematical one. @Nishant: It seems reasonable to me to ask a student to reduce an integral to a table entry, then specify the entry that is appropriate. – Ross Millikan Nov 21 '14 at 03:42
  • @Nishant: (continued) that shows a skill-to recognize when the integral is in a form that matches the table entry. – Ross Millikan Nov 21 '14 at 03:44

1 Answers1

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Hint: If $$\int (x-3)\sqrt{6+6x-x^2}\, dx $$ take $u=6+6x-x^2\implies du=(6-2x)dx=-2(x-3)dx$ then

$$\int (x-3)\sqrt{6+6x-x^2}\, dx=\frac{-1}{2}\int u^{1/2} du $$

AsdrubalBeltran
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  • Yes actually there is a choice for u-sub for that....I didn't include it here as I thought that wouldn't be right. Guess I'm wrong lol...damnnnnn – Zhi J Teoh Nov 21 '14 at 03:31