# another integral

• Oct 27th 2006, 10:12 PM
scorpion007
another integral
Find an antiderivative of $\frac{2+6x}{\sqrt{4-x^2}}$

Now I know it has something to do with arcsin, but I'm not exactly sure what i can do to clean up that numerator...
• Oct 27th 2006, 10:54 PM
CaptainBlack
Quote:

Originally Posted by scorpion007
Find an antiderivative of $\frac{2+6x}{\sqrt{4-x^2}}$

Now I know it has something to do with arcsin, but I'm not exactly sure what i can do to clean up that numerator...

Note that:

$\frac{d}{dx} \sqrt{4-x^2} = \frac{-x}{\sqrt{4-x^2}}$.

So if we rewrite the problem as find the antiderivative of:

$\frac{2+6x}{\sqrt{4-x^2}} = \frac{2}{\sqrt{4-x^2}} + \frac{6x}{\sqrt{4-x^2}}$

as the first term on the right is a standard integral, and the second is $-6$ times what we had earlier you should be able to complete this from here.

RonL
• Oct 27th 2006, 11:13 PM
scorpion007
is this correct?
$2\arcsin(\frac{x}{2}) - 6\sqrt{4-x^2}$
• Oct 28th 2006, 12:39 AM
CaptainBlack
Quote:

Originally Posted by scorpion007
is this correct?
$2\arcsin(\frac{x}{2}) - 6\sqrt{4-x^2}$

Looks OK to me.

Checking by differentiation is shown in the attachment.

RonL
• Oct 28th 2006, 07:22 AM
Soroban
Hello, scorpion007!

This is the Captain's solution . . . in baby-talk.

Quote:

$\int \frac{2+6x}{\sqrt{4-x^2}}\,dx$

We have: . $\int\frac{2+6x}{\sqrt{4-x^2}}\,dx \;=\;\int\left(\frac{2}{\sqrt{4-x^2}} + \frac{6x}{\sqrt{4-x^2}}\right)dx
$

. . . . . $=\;2\!\int\!\!\frac{dx}{\sqrt{4-x^2}} \;+ \;6\!\int\!\! x(4-x^2)^{-\frac{1}{2}}dx$

The first integral is of the $\arcsin$ form.

For the second integral, let $u = 4-x^2$

• Oct 28th 2006, 07:44 PM
scorpion007
thanks, i understand perfectly.