Integration of Rational Functions by partial fractions

**FalconPUNCH!** · February 21st 2008, 03:02 PM

I understand how to do it but I get stuck really easily. Like this for example

$\int_{0}^{2\sqrt{3}} \frac{x^3}{\sqrt{16-x^2}}$

Yeah it's probably an easy one but I'm not that great with limit of integration. so I get

$x = 4sin(o)$
$dx = 4cos(o)$

So here's where I am stuck. I don't know what to do now. I plug those in but I don't know what my limit of integration is supposed to be. This stuff is killing me. I'm trying really hard to understand. Please someone help me.

**Krizalid** · February 21st 2008, 03:08 PM

Originally Posted by FalconPUNCH!

I understand how to do it but I get stuck really easily. Like this for example

$\int_{0}^{2\sqrt{3}} \frac{x^3}{\sqrt{16-x^2}}$

Much better to substitute $u^2=16-x^2.$

Originally Posted by FalconPUNCH!

$x = 4sin(o)$

For $x=0\implies 0=4\sin\theta.$ For $x=2\sqrt3\implies 2\sqrt3=4\sin\theta.$ Compute $\theta$ in both cases.

**FalconPUNCH!** · February 21st 2008, 03:13 PM

Originally Posted by Krizalid

Much better to substitute $u^2=16-x^2.$

For $x=0\implies 0=4\sin\theta.$ For $x=2\sqrt3\implies 2\sqrt3=4\sin\theta.$ Compute $\theta$ in both cases.

So in order to solve for what the new limits of integration I have to set what I'm substituting like in this example sin(o) = to the old limits of integration and solve. Thanks Krizalid.

**galactus** · February 21st 2008, 03:17 PM

If you must use trig sub, you have the correct subs.

Did you sub them in and simplify?. Also, you have to change your limits of integration because you're now in terms of theta.

$x=4sin({\theta}), \;\ dx=4cos({\theta})d{\theta}$

Therefore, solve for theta using our old limits.

$2\sqrt{3}=4sin({\theta}); \;\ {\theta}=\frac{\pi}{3}$

$\int\frac{64sin^{3}({\theta})}{\sqrt{16-(4sin({\theta}))^{2}}}\cdot{4cos({\theta})}d{\thet a}$

$=64\int_{0}^{\frac{\pi}{3}}sin^{3}({\theta})d{\the ta}$

Can you finish?.

**Soroban** · February 21st 2008, 03:38 PM

Hello, FalconPUNCH!

$\int_{0}^{2\sqrt{3}} \frac{x^3}{\sqrt{16-x^2}}\,dx$

You said: . $x \:= \:4\sin\theta\quad\Rightarrow\quad dx \:= \:4\cos\theta\,d\theta$

. . Substitute: . $\int\frac{64\sin^3\!\theta}{4\cos\theta}(4\cos\the ta\,d\theta) \;=\;64\!\!\int\!\sin^3\!\theta\,d\theta$

To change the limits, recall your substitution:
. . . $x \:=\:4\sin\theta \quad\Rightarrow\quad \sin\theta \:=\:\frac{x}{4}$

When $x = 0$ , we have: . $\sin\theta \:=\:\frac{0}{4} \:=\:0\quad\Rightarrow\quad\boxed{\theta \:=\:0}$

When $x = 2\sqrt{3}$ , we have: . $\sin\theta \:=\:\frac{2\sqrt{3}}{4} \:=\:\frac{\sqrt{3}}{2}\quad\Rightarrow\quad\boxed {\theta \:=\:\frac{\pi}{3}}$

. . And there are the new limits . . .

The integral becomes: . $64\!\int^{\frac{\pi}{3}}_0\!\!\sin^3\!\theta\,d\th eta$

Got it?

**FalconPUNCH!** · February 21st 2008, 03:39 PM

Originally Posted by galactus

If you must use trig sub, you have the correct subs.

Did you sub them in and simplify?. Also, you have to change your limits of integration because you're now in terms of theta.

$x=4sin({\theta}), \;\ dx=4cos({\theta})d{\theta}$

Therefore, solve for theta using our old limits.

$2\sqrt{3}=4sin({\theta}); \;\ {\theta}=\frac{\pi}{3}$

$\int\frac{64sin^{3}({\theta})}{\sqrt{16-(4sin({\theta}))^{2}}}\cdot{4cos({\theta})}d{\thet a}$

$=64\int_{0}^{\frac{\pi}{3}}sin^{3}({\theta})d{\the ta}$

Can you finish?.

I got up to the last part the only thing I was having trouble with was the limits of integration.

**FalconPUNCH!** · February 21st 2008, 03:40 PM

Originally Posted by FalconPUNCH!

I got up to the last part the only thing I was having trouble with was the limits of integration.

Originally Posted by Soroban

Hello, FalconPUNCH!

You said: . $x \:= \:4\sin\theta\quad\Rightarrow\quad dx \:= \:4\cos\theta\,d\theta$

. . Substitute: . $\int\frac{64\sin^3\!\theta}{4\cos\theta}(4\cos\the ta\,d\theta) \;=\;64\!\!\int\!\sin^3\!\theta\,d\theta$

To change the limits, recall your substitution:
. . . $x \:=\:4\sin\theta \quad\Rightarrow\quad \sin\theta \:=\:\frac{x}{4}$

When $x = 0$ , we have: . $\sin\theta \:=\:\frac{0}{4} \:=\:0\quad\Rightarrow\quad\boxed{\theta \:=\:0}$

When $x = 2\sqrt{3}$ , we have: . $\sin\theta \:=\:\frac{2\sqrt{3}}{4} \:=\:\frac{\sqrt{3}}{2}\quad\Rightarrow\quad\boxed {\theta \:=\:\frac{\pi}{3}}$

. . And there are the new limits . . .

The integral becomes: . $64\!\int^{\frac{\pi}{3}}_0\!\!\sin^3\!\theta\,d\th eta$

Got it?

yeah it's pretty clear now. Thanks guys

**FalconPUNCH!** · February 21st 2008, 03:46 PM

So the answer I get for this is $\frac{24\sqrt{3} - 128}{3}$

**galactus** · February 21st 2008, 03:52 PM

No, I am sorry to say, that is incorrect.

**xifentoozlerix** · February 21st 2008, 04:13 PM

You really shouldn't let the limits of integration stop you from solving the problem. If you are having trouble finding the new limits, simply forget about them until you have to evaluate your $u$ function. Then just "un-substitute" your original function back in place of $u$ and use the original limits.

**FalconPUNCH!** · February 21st 2008, 04:16 PM

I'm really stressing out over this. I can't even solve a single integral because of this

I'm trying to do the ones ahead but they're even more annoying. I might do what you said, xifentoozlerix but I get confused when I reach the end.

Here's what I'm doing:

$64\int_{0}^{\frac{pi}{3}} sin^{2}0 sin(0) d0$

$u = cos(o), du = -sin(o)do$

$64\int_{0}^{\frac{pi}{3}} (1-cos^{2}x) sin(0) d0$

$64\int_{0}^{\frac{pi}{3}} (1 - u^2) -du$

$-64( u - \frac{1}{3}u^{3})$

$-64( cos(o) - \frac{1}{3}cos(o)^{3})$

I then plugged in pi/3 and 0 and the answer I posted before is what I got. I'm not sure where I messed up.

**galactus** · February 21st 2008, 04:30 PM

Perhaps try the advice that was given about waiting 'til the end.

You can always use the reduction formula for sine. Save you all the grunt work.

$\int{sin^{n}(u)}du=\frac{-1}{n}sin^{n-1}(u)cos(u)+\frac{n-1}{n}\int{sin^{n-2}}du$

**FalconPUNCH!** · February 21st 2008, 04:42 PM

Okay I haven't used to reduction formula a lot so I'm not sure if I did it correctly

$64( \frac{-1}{3}sin^{2}(u)cos(u) + \frac{2}{3} \int sin du)$

$64( \frac{-1}{3}sin^{2}(\frac{pi}{3})cos(\frac{pi}{3}) - sin^{2}(0)cos(0) + \frac{2}{3} ( \frac{1}{2} - \frac{pi}{2} )$

$64(( (\frac{-1}{3}) (\frac{3}{4})(\frac{1}{2}) - \frac{pi}{2}) + \frac{2}{3} (\frac{1}{2} - \frac{pi}{2})$

I then simplify it to

$64( \frac{-8 - 23pi}{24})$

not sure if it's right...

**mr fantastic** · February 21st 2008, 04:50 PM

Originally Posted by FalconPUNCH!

Okay I haven't used to reduction formula a lot so I'm not sure if I did it correctly

$64( \frac{-1}{3}sin^{2}(u)cos(u) + \frac{2}{3} \int sin du)$

$64( \frac{-1}{3}sin^{2}(\frac{pi}{3})cos(\frac{pi}{3}) - sin^{2}(0)cos(0) + \frac{2}{3} ( \frac{1}{2} - \frac{pi}{2} )$

$64(( (\frac{-1}{3}) (\frac{3}{4})(\frac{1}{2}) - \frac{pi}{2}) + \frac{2}{3} (\frac{1}{2} - \frac{pi}{2})$

I then simplify it to

$64( \frac{-8 - 23pi}{24})$

not sure if it's right...

$I = 64 \int_{0}^{\pi/3} \sin^3 \theta \, d\theta = 64 \int_{0}^{\pi/3} \sin^2 \theta \, \sin \theta \, d\theta = 64 \int_{0}^{\pi/3} (1 - \cos^2 \theta) \sin \theta \, d\theta$ .

Substitute $u = \cos \theta$ . Then:

$I = -64 \int_{1}^{1/2} 1 - u^2 \, du = ......$

I get $I = \frac{40}{3}$ .

**FalconPUNCH!** · February 21st 2008, 04:56 PM

Originally Posted by mr fantastic

$I = 64 \int_{0}^{\pi/3} \sin^3 \theta \, d\theta = 64 \int_{0}^{\pi/3} \sin^2 \theta \, \sin \theta \, d\theta = 64 \int_{0}^{\pi/3} (1 - \cos^2 \theta) \sin \theta \, d\theta$ .

Substitute $u = \cos \theta$ . Then:

$I = -64 \int_{1}^{1/2} 1 - u^2 \, du = ......$

I get $\frac{40}{3}$

hmmm...seem to be getting the limits of integration wrong which is happening in my homework a lot.

for this one $\int_{1}^{2} \frac{\sqrt{x^{2} - 1}}{x}dx$ is it best to use u substitution from the start?

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