1. ## Some integration problems

hi

I am having a bit of trouble on the following integration problems and was wondering if anybody could help me out.

1) Integrate: ( x^2 ) / sqrt(9- x^2 ) dx

I think this one deals with trig substitution, but I am unsure of the exact method

2) Integrate: dx/(x^2 * sqrt(1 + x^2 ))

3) Integrate: x^2 / ( 25 + x ^2 ) dx

with 2 & 3, I'm really not sure of the method to start them. any help is appreciated; thanks alot!

2. Originally Posted by ChaosBlue
1) Integrate: ( x^2 ) / sqrt(9- x^2 ) dx

I think this one deals with trig substitution, but I am unsure of the exact method
First let 3y = x. Then y = (1/3)x and dy = (1/3)dx so:
Int[x^2/sqrt(9 - x^2)dx = Int[(3y)^2/sqrt(9 - (3y)^2)*3dy] = 9*Int[y^2/sqrt(1 - y^2)dy]

Now let y = sin(t). Then dy = cos(t) dt. Thus:
Int[x^2/sqrt(9 - x^2)dx = 9*Int[y^2/sqrt(1 - y^2)dy] = 9*Int[sin^2(t)/sqrt(1 - sin^2(t))*cos(t)dt]

= 9*Int[sin^2(t)/cos(t)*cos(t)dt] = 9*Int[sin^2(t)dt]

Now you take it from here.

-Dan

3. Originally Posted by ChaosBlue
2) Integrate: dx/(x^2 * sqrt(1 + x^2 ))
Same idea as 1) but with a different trig substitution:
Let x = tan(t). Then dx = sec^2(t). Thus:
Int[x^2/sqrt(1 + x^2)dx] = Int[tan^2(t)/sqrt(1 + tan^2(t))*sec^2(t)dt]

Now, 1 + tan^2(t) = sec^2(t):

Int[x^2/sqrt(1 + x^2)dx] = Int[tan^2(t)/sqrt(1 + tan^2(t))*sec^2(t)dt] = Int[tan^2(t)/sec(t)*sec^2(t)dt]

= Int[sec(t)*tan^2(t)dt] = Int[sin^2(t)/cos^3(t) dt]

Now you take it from here.

-Dan

4. Originally Posted by ChaosBlue
3) Integrate: x^2 / ( 25 + x ^2 ) dx
This is a mix between 1) and 2):
First let 5y = x. Then x =(1/5)y and dx = (1/5)dy. Thus:
Int[x^2/(25 + x^2) dx] = Int[(5y)^2/(25 + (5y)^2) * 5dy] = 5*Int[y^2/(1 + y^2) dy]

Now let y = tan(t). Take it from here.

-Dan

5. Originally Posted by ChaosBlue
hi

I am having a bit of trouble on the following integration problems and was wondering if anybody could help me out.

1) Integrate: ( x^2 ) / sqrt(9- x^2 ) dx

I think this one deals with trig substitution, but I am unsure of the exact method

2) Integrate: dx/(x^2 * sqrt(1 + x^2 ))

3) Integrate: x^2 / ( 25 + x ^2 ) dx

with 2 & 3, I'm really not sure of the method to start them. any help is appreciated; thanks alot!

Hello,

Here are the answers. They are all in order just click on the links and follow the steps. I know that some of them seem rather long, but in most of the cases I am just arrowing over with my calculator!

1. Integrate: ( x^2 ) / sqrt(9- x^2 ) dx

a. http://item.slide.com/r/1/114/i/Jxlz...Ljm6JoXZpvx5_/
b. http://item.slide.com/r/1/168/i/Uu3z...OiZevEggkYso1/
c. http://item.slide.com/r/1/104/i/o92O...V8xbT3_HltsdQ/
d. http://item.slide.com/r/1/27/i/yIm0M...UXOqjulE05xAI/
e. http://item.slide.com/r/1/23/i/SoQPx...g8hvAKCWf9G9I/

2. Integrate: dx/(x^2 * sqrt(1 + x^2 ))

a. http://item.slide.com/r/1/73/i/5oJLP...KdNNK5OsRn_9i/
b. http://item.slide.com/r/1/79/i/KfHZF...UzhT3uV-MbaS7/
c. http://item.slide.com/r/1/93/i/V6afN...S8tEmt7eXPToE/
d. http://item.slide.com/r/1/139/i/aVVZ...kby9vaZKgd2fW/
f. http://item.slide.com/r/1/114/i/7byb...zlporHYxZ4dUq/
g. http://item.slide.com/r/1/51/i/5TMiq...xjZX7002bZSDm/
h. http://item.slide.com/r/1/125/i/TY_G...Ct4frI5KXyoa8/
i. http://item.slide.com/r/1/11/i/-1jHG...LY7V7YCyGmkw1/
j. http://item.slide.com/r/1/159/i/mlDT...ZpehScT-26Kmb/
k. http://item.slide.com/r/1/49/i/pHBDW...hgFuu1qiODc1T/
l. http://item.slide.com/r/1/32/i/V29Ji...G4P6KQBW2NQTa/
m. http://item.slide.com/r/1/101/i/eQCg...F6dQJz4zg7Zlr/
n. http://item.slide.com/r/1/68/i/MPTar...5oMpn_9YHMHcq/
o. http://item.slide.com/r/1/14/i/Rkigt...Qo-vhs5hFxUer/
p. http://item.slide.com/r/1/162/i/NE8a...h9Zj-XCdNhgyJ/
q. http://item.slide.com/r/1/21/i/3Jfju...yj5yRRr3aljDI/
r. http://item.slide.com/r/1/21/i/-nsRc...PKzZpy86gwC9P/

2. Integrate: x^2 / ( 25 + x ^2 ) dx

a. http://item.slide.com/r/1/151/i/A3f_...rdKR3g9yXColQ/
b. http://item.slide.com/r/1/151/i/dlYk...1la8WyLu-DFLT/
c. http://item.slide.com/r/1/168/i/ZAqK...m8WTt6KnrGD78/
d. http://item.slide.com/r/1/152/i/4ZB9...wHDd3gN109GBc/

There you go now were done!

6. Hello, ChaosBlue!

All of them deal with Trig Substitution . . .

. . . . . x² dx
1) . ∫ ---------
. . . . √9 - x²

Let: x = 3·sinθ . . dx = 3·cosθ·dθ
. . and the radical becomes: 3·cosθ

. . . . . . . . . . .3·sin²θ
Substitute: . ∫ --------- (3·cosθ·dθ) . = . 3 ∫ sin²θ dθ
. . . . . . . . . . .3·cosθ

Can you finish it now?

. . . . . . . dx
2) . ∫ --------------
. . . . .x²√1 + x²

Let x = tanθ . . dx = sec²θ·dθ
. . and the radical becomes: secθ

. . . . . . . . . . . sec²θ·dθ . . . . . . .cosθ
Substitute: . ∫ ------------- . = . ∫ ------- . = . ∫ cscθ·cotθ dθ . . . etc.
. . . . . . . . . . .tan²θ·secθ . . . . . .sin²θ

. . . . . .x² dx
3) . ∫ ----------
. . . . .25 + x²

Let: x = 5·tanθ . . dx = 5·sec²θ·dθ
. . The denominator becomes: 25·sec²θ

. . . . . . . . . . . 25·tan²θ
Substitute: . ∫ ------------ (5·sec²θ·dθ) . = . 5 ∫ tan²θ dθ . = . 5 ∫ (sec²θ - 1) dθ
. . . . . . . . . . . 25·sec²θ

Got it?