Question

When using u-substitution in integrating, we can't just introduce "new variables, right? so, like, integrating (sin x)^3, the u=sinx, and the du=cosx dx, and so if I were to use the u substitution, I would have to say (cosx)^-1 int(u^3)...would that work or no?

Answer 1

$$ \int sin^3(x) $$
is a tough integration.

Here is a video explaining it: https://www.youtube.com/watch?v=VeULk9R1z5E

Answer 2

I had trouble with it when I started calc 2.

Answer 3

Understand when you're making a substitution, you're replacing one variable with another.

So yes, you can do something like u=sin x.
You just can't replace your variable with the same variable: like x = sin x.
Err maybe you could .. whatev.. that's not how we treat subs though XD

Unfortunately u=sin x won't help in that particular problem.
Oh oh we have an ODD power on sine, so this problem works out really nicely.

Answer 4

\[\Large\rm \int\limits \sin^3x~dx=\int\limits (1-\cos^2x)\sin x~ dx\]Make the substitution u=cos x.

Answer 5

^ you get this from the trig identity
$$ sin^2(x) + cos^2(x) = 1 $$
$$ \therefore sin^2(x) = 1 - cos^2(x) $$

Answer 6

OMAGOD THAT IS AMAZING

Answer 7

but ok so introducing another x is NOT okay, right?

Answer 8

You'll need to know your trig identities! Brush up if you are unfamiliar with them. You use them in ~half of Calc integrals.

Answer 9

Well, I'm not sure if it's "legal" or not in this sense, but it's definitely not helpful. You don't want to introduce variables if they only make the integral harder.

Answer 10

Yah don't introduce another x :U silly

Answer 11

These integrals can be very easy or quite burdensome depending on what the integral looks like.

For example:\[\Large\rm \int\limits \sin^{51}x~dx\]Would be MUCH easier to solve than,\[\Large\rm \int\limits \sin^6x~dx\]Even powers require a bit of fancy footwork.

Answer 12

0-0 WATTTT a power of 51 is easier????

Answer 13

hehe yah XD it's a simple u-sub.

The power of 6 .. you need to either apply the Half-Angle Identity for Sine like ... 5 times,
Or use the Sine Reduction Formula 3 times.

Answer 14

To finish the integral, you would let u = cos(x).
Therefore, du = -sin(x)dx -> which can be rewritten as -> -du = sin(x)dx

And now do:

$$ \int (1- u^2) -du $$

Pull the negative out (it's just a constant)
$$ -\int (1- u^2) du $$

Distribute the du
$$ -\int du + u^2 du $$

Split into two integrals
$$ -[\int du + \int u^2 du] $$

Solve the integrals in terms of u
$$ -[u + \frac{u^3}{3}] $$

Plug cos back in
$$ -(cos(x)) + \frac{(cos(x))^3}{3} $$

Which can also be rearranged as
$$\frac{(cos(x))^3}{3} -(cos(x)) $$

Answer 15

Just some motivation, I was failing Calc 2 at the beginning, I didn't get any of this.

STUDY AND PRACTICE!!! I worked my butt off, ended up with an A, and I should get an A in Calc 3. It's trivial once you run through a bunch of examples. Practice, practice, practice, makes perfect :)

Answer 16

0-0 woah. so then the sin^51 is basically the same except a whole bunch of ... (1−u2)?

Answer 17

hmmm LOL I have the AP exam wednesday ( that's tomorrow) and I haven't gotten too in-depth with these... i'm not feeling too good right now...

Answer 18

Should I do it all tomorrow...sigh I have a 4 chemistry reports to turn in the very same day, so might as well do that... thanks, though!!! Ya'll are SUPER AWESOME

Answer 19

for the 51st power?
Break it up into 50 and 1,\[\Large\rm \int\limits \sin^{50}x \sin x~dx\]Then rewrite your 50 powers using the square identity,\[\Large\rm \int (\sin^{2}x)^{25}\sin x~dx\]\[\Large\rm \int\limits (1-\cos^2x)^{25}\sin x~ dx\]Then again we do our u-sub! :)
u=cos x

Answer 20

Chemistry? :O oh boy.

Answer 21

DANG that is a very powerful property...and yes...chemistry...nbfhleugwiahokdsnbvjhfidsajnkcvbhf I am SOOOO done with chemistry

Answer 22

XD

Answer 23

any tips, perhaps, for chemistry? I have a final tomorrow...

Answer 24

Hmm naw :c Chem was a little tough for me.

I like the equations where stuff turns into other stuff and you gotta balance it.
Those feel more math'y.

But overall, chem felt like too much memorization >.<

Answer 25

hahahahah LOL stuff turns into other stuffz LOL so true~ well, thanks ^^ I'll try to keep that in mind...stuffz=>other schtuffz HAHAHA

Answer 26

XD