Another seemingly simple integral

**fattydq** · February 25th 2009, 01:02 PM

The definite integral from -1/2 to 1/2 of 9/the square root of 1-x^2 dx

I don't even know where to begin in this one. Finding the antiderivative of that seems impossible...

**Krizalid** · February 25th 2009, 01:05 PM

$\int_{-1/2}^{1/2}{\frac{9}{\sqrt{1-x^{2}}}\,dx}=2\int_{0}^{1/2}{\frac{9}{\sqrt{1-x^{2}}}\,dx}.$

Look at a table of integrals, that primitive is the arcsin function.

**fattydq** · February 25th 2009, 01:10 PM

Originally Posted by Krizalid

$\int_{-1/2}^{1/2}{\frac{9}{\sqrt{1-x^{2}}}\,dx}=2\int_{0}^{1/2}{\frac{9}{\sqrt{1-x^{2}}}\,dx}.$

Look at a table of integrals, that primitive is the arcsin function.

I know that, but what did you just do knowing that? Where did the 2 come from and why did the -1/2 just disappear? And I know that 1/the sq root of 1-x^2 but it's a 9 not a 1, so where does this come into play?

**Krizalid** · February 25th 2009, 01:22 PM

$\int_{-a}^af=2\int_0^af$ provided that $f$ is an even function and $a\in\mathbb R.$

**Abu-Khalil** · February 25th 2009, 01:26 PM

Originally Posted by fattydq

I know that, but what did you just do knowing that? Where did the 2 come from and why did the -1/2 just disappear? And I know that 1/the sq root of 1-x^2 but it's a 9 not a 1, so where does this come into play?

$\int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx=<br /> \int_{-\frac{1}{2}}^{0}\frac{9}{\sqrt{1-x^2}}dx+\int_{0}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx$ ,
but $\frac{9}{\sqrt{1-(-x)^2}}=\frac{9}{1-x^2}\Rightarrow \int_{-\frac{1}{2}}^{0}\frac{9}{\sqrt{1-x^2}}dx=\int_{0}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx$ .
Therefore $\int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx=<br /> 2\int_{0}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx=18\int_{0}^{\frac{1}{2}}\frac{dx}{\sqrt{1-x^2}}$ .

PD: Late >.<

**fattydq** · February 25th 2009, 01:44 PM

Originally Posted by Krizalid

$\int_{-a}^af=2\int_0^af$ provided that $f$ is an even function and $a\in\mathbb R.$

I love calculus, because there's rule's for everything that I'm just supposed to know magically, intuitively. I swear my professor never mentioned anything like that.

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