Question

integral((x^2)/(4-3x^2)^(3/2))dx

Answer 1

\[\int\limits \frac{ x^2 }{ (4-3x^2)^(3/2) }dx\]

Answer 2

Trignometric substitution with a difference of squares

Answer 3

\[(\sqrt{4})^2-3x^2\]
\[x=\frac{ 2 }{ 3 } \sin(theta)\]

Answer 4

well take x^2 common from denominator

Answer 5

can we use a u-sub instead of trig sub

Answer 6

\[u=4-3x^2\]
\[du=-6xdx\]
\[dx=-(1/6x)dx\]

Answer 7

no i would stick with the trig sub, with a minor correction
\[x = \frac{2}{\sqrt{3}} \sin \theta\]

Answer 8

oh, ok

Answer 9

so would i plug that into x for the whole equation including the top (x^2)

Answer 10

yes and the dx as well
\[dx = \frac{2}{\sqrt{3}} \cos \theta \]

Answer 11

it should simplify to
\[\frac{1}{3 \sqrt{3}} \int\limits \tan^2 \theta\]

Answer 12

is that for the top part?

Answer 13

no the whole thing

top part ---> sin^2
bottom part ----> cos^3
dx ----> cos

tan = sin/cos

Answer 14

ok, but what do you do with the rest of the equation?
wolfram gives we this complex number
http://www4a.wolframalpha.com/Calculate/MSP/MSP1071222f045f1f7caf0b00001g1fh4b1da7ai778?MSPStoreType=image/gif&s=12&w=443.&h=57.

Answer 15

well first do you see how it simplifies to tan^2 ?

Answer 16

yes

Answer 17

ok, now we can use the identity
\[\tan^2 \theta = \sec^2 \theta - 1\]
this makes the integration much easier
\[\frac{1}{3 \sqrt{3}} \int\limits \sec^2 \theta - 1 = \frac{1}{3 \sqrt{3}}(\tan \theta - \theta)\]

Answer 18

you follow?

Answer 19

I think so hold on.

Answer 20

ok ... derivative of tan = sec^2 is a known trig identity that is why the integrating was easy

Answer 21

what do you do with \[\frac{ 1 }{ 3\sqrt{3} }\tan(\theta)-\theta+c\]

Answer 22

Finally we have to plug "x" back in

from original substitution
\[x = \frac{2}{\sqrt{3}} \sin \theta\]
\[\rightarrow \sin \theta = \frac{\sqrt{3}}{2}x\]
\[\rightarrow \theta = \sin^{-1} (\frac{\sqrt{3}}{2}x)\]
For the tan part, it helps to use a triangle
remember SOH CAH TOA

\[\rightarrow \tan \theta = \frac{\sqrt{3} x}{\sqrt{4-3x^2}}\]

Answer 23

thank you~ I got the same answer as wolfram!!!

Answer 24

yw :)

sometimes you may not get exactly same as wolfram because they factor or simplify differently but it doesn't necessarily mean you are wrong