Solving Integral of (9-4x^2)^1/2 with Substitution Method

[Phil Frehz]

Hey everyone, I'm currently studying for Calc 3 and came across this integral that his been racking my brain beyond insanity. I know the solution is easier than it is. I have looked online and come across substituting to have x=2sin(theta).

I also came across a step where you substitute u for 2x leaving you with:

integral of (25-u²)^1/2

I feel like there is a simpler way to solve it but I can't seem to see it

Any help will greatly be appreciated.

 

[Chestermiller]

This problem begs to be solved by trig substitution. Try $x=\frac{3}{2}\sinθ$, and then see if you can figure out why it works so well.

Chet

 

[Qwertywerty]

Substitute x= (3/2)×sin(θ) .

 

[SteamKing]

View attachment 86106 Hey everyone, I'm currently studying for Calc 3 and came across this integral that his been racking my brain beyond insanity. I know the solution is easier than it is. I have looked online and come across substituting to have x=2sin(theta).

I also came across a step where you substitute u for 2x leaving you with:

integral of (25-u²)^1/2

I feel like there is a simpler way to solve it but I can't seem to see it

Any help will greatly be appreciated.

It's not clear that making the substitution u = 2x into (9 - 4x²)^1/2 leads to (25 - u²)^1/2.

Have you tried this yourself and worked out the algebra to confirm?

In any event, dust off your trig identity knowledge. I'm not saying that the substitution x = 2 sin θ is correct here, but trig substitution is one way to go.

 

[Phil Frehz]

Alright I posted some of the work but I'm having trouble with the dx and d(theta).

 

[Qwertywerty]

You haven't substituted correctly for dx .

 

[Qwertywerty]

Also you need to now change the limits of your integral .

 

[Phil Frehz]

Forgot to change the limits but I think I correctly included dx as d(theta)

 

[Qwertywerty]

You are supposed to substitute dx by (3/2)×cos(θ).dθ

 

[Phil Frehz]

Sorry added the wrong pic

 

[Qwertywerty]

So now just change your limits and you will get the answer .

 

[Phil Frehz]

Alright great got it! Thanks for the help everyone.