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Am I correct for pattern matching this integral?

I have

$$\int \frac{\sqrt{9x^2+4}}{x^2}dx$$

Does this pattern match with:

$$\int \frac{\sqrt{a^2 + x^2}}{x^2}dx = -\frac{a^2 + x^2}{x} + \ln(x + \sqrt{a^2 + x^2}) + c$$

If I factor out the 9, I get

$$= 3 \int \frac{\sqrt{x^2 + \frac{4}{9}}}{x^2}$$ with $a = \frac{2}{3}$

I get: $$3 \left( - \frac{\sqrt{\frac{4}{9}+x^2}}{x} + \ln\left(x+\sqrt{\frac{4}{9}+x^2}\right) +c\right)$$

Is this the right track?

Wolfram winds up with a different answer though:

enter image description here

clathratus
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Jwan622
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    Have you tried manipulating your answer to look like Wolfram's? Also, note that it's fine if your answer differs from Wolfram's by a constant. – Minus One-Twelfth Mar 04 '19 at 01:58
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    Actually they are the same, because the answer given by WA is simply yours plus one constant $3 \log (3/2)$. – xbh Mar 04 '19 at 02:01
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    One quick way to check the result is to put your formula into Wolfram Alpha (without the constant, which in this case is just adding $3c$ to the result) and subtract the formula Wolfram Alpha gave (again omitting the constant). If the result is a flat constant function then your integral is correct. – David K Mar 04 '19 at 04:18

1 Answers1

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You are indeed correct. Note that in an indefinite integral, perfectly valid answers can be separated by (any) constant.

The first term in Wolfram's answer can be rewritten:

$3\ln{(\frac32(x+\sqrt{\frac49 + x^2}))} = 3\ln{(x+\sqrt{\frac49 + x^2})} + 3\ln\frac 32$

and the second term can be rearranged to be identical to your other term.

So your answers are separated by a constant. That's fine. You're right.

Deepak
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