Integrating Using Trigonometric Substitution

[prosteve037]

Homework Statement

Use the method of trigonometric substitution to evaluate the following:


$\int\frac{x^{2}}{\sqrt{4-9x^{2}}}$

Homework Equations

The only relevant equation that I could think of for this one was the trig identity:

$sin^{2}\vartheta + cos^{2}\vartheta = 1$

The Attempt at a Solution

This is what I got from trying this problem:

$\int\frac{x^{2}}{\sqrt{4-9x^{2}}}$

$\sqrt{4-9x^{2}}$ is a difference between two squares, so it's a leg of a right triangle. So I let $4sin\vartheta = \sqrt{4-9x^{2}}$ to satisfy the definition of sine (Opposite Side over Hypotenuse).

Then, I let $x^{2} = \frac{4}{9}cos\vartheta$ to satisfy the definition of cosine.

Differentiating this same function with respect to $x$, I got:

$\frac{\frac{-2}{9}sin\vartheta}{\sqrt{\frac{4}{9}cos\vartheta}}$

Substituting these into the original function:

$\int{\frac{\frac{4}{9}cos\vartheta}{4sin\vartheta} \times \frac{\frac{-2}{9}sin\vartheta}{\sqrt{\frac{4}{9}cos\vartheta}}d\vartheta}$

$\int{\frac{\frac{4}{9}cos\vartheta}{4sin\vartheta} \times \frac{\frac{-2}{9}sin\vartheta}{\frac{2}{3}\sqrt{cos\vartheta}}d\vartheta}$

$\int{\frac{1}{9} \times \frac{cos\vartheta}{sin\vartheta} \times \frac{-1}{3} \times \frac{sin\vartheta}{\sqrt{cos\vartheta}}d\vartheta}$

$\frac{-1}{27}\int{\frac{cos\vartheta}{\sqrt{cos\vartheta}}d\vartheta}$

At this point, I got stuck and didn't know what to do. I tried using a modified version of the identity above and just went on to see if I could get anywhere but failed.

Thanks

 

[GreenPrint]

they hypotenuse is actually 2, remember c^2=a^2 + b^2, the other leg is actually 3x

 

[Bohrok]

$\sqrt{4-9x^{2}}$ is a difference between two squares, so it's a leg of a right triangle. So I let $4sin\vartheta = \sqrt{4-9x^{2}}$ to satisfy the definition of sine (Opposite Side over Hypotenuse).

Then, I let $x^{2} = \frac{4}{9}cos\vartheta$ to satisfy the definition of cosine.

I can't really see how you got these substitutions... A standard substitution for this problem would be $x = \frac{2}{3}\cos\vartheta$

 

[SammyS]

Homework Statement

Use the method of trigonometric substitution to evaluate the following:


$\int\frac{x^{2}}{\sqrt{4-9x^{2}}}$

Homework Equations

The only relevant equation that I could think of for this one was the trig identity:

$sin^{2}\vartheta + cos^{2}\vartheta = 1$

The Attempt at a Solution

This is what I got from trying this problem:

$\int\frac{x^{2}}{\sqrt{4-9x^{2}}}$

$\sqrt{4-9x^{2}}$ is a difference between two squares, so it's a leg of a right triangle. So I let $4sin\vartheta = \sqrt{4-9x^{2}}$ to satisfy the definition of sine (Opposite Side over Hypotenuse).

...

It may help you to indicate that x is the variable of integration, by including dx in your integral: $\displaystyle \int\frac{x^{2}}{\sqrt{4-9x^{2}}}\,dx\,.$

To elaborate on GreenPrint's post:

If $\sqrt{4-9x^{2}}$ is a leg of a right triangle, then the hypotenuse would likely be 2 and the other leg 3x.

As you say, sine is opposite over hypotenuse, so either $\displaystyle \sin(\theta)=\frac{3x}{2}$ or $\displaystyle \sin(\theta)=\frac{\sqrt{4-9x^{2}}}{2}\,.$

cos(θ) is the other.

 

[Harrisonized]

Just making info easier to understand.

[PLAIN]http://img856.imageshack.us/img856/4011/derpto.png

Make sure you find dx to get dθ.

 

[HallsofIvy]

Homework Statement

Use the method of trigonometric substitution to evaluate the following:


$\int\frac{x^{2}}{\sqrt{4-9x^{2}}}$

Homework Equations

The only relevant equation that I could think of for this one was the trig identity:

$sin^{2}\vartheta + cos^{2}\vartheta = 1$

The Attempt at a Solution

This is what I got from trying this problem:

$\int\frac{x^{2}}{\sqrt{4-9x^{2}}}$

$\sqrt{4-9x^{2}}$ is a difference between two squares, so it's a leg of a right triangle. So I let $4sin\vartheta = \sqrt{4-9x^{2}}$ to satisfy the definition of sine (Opposite Side over Hypotenuse).

Then, I let $x^{2} = \frac{4}{9}cos\vartheta$ to satisfy the definition of cosine.

No, you must have [itex]x= (2/9)cos(\vartheta)[/tex].

Differentiating this same function with respect to $x$, I got:

$\frac{\frac{-2}{9}sin\vartheta}{\sqrt{\frac{4}{9}cos\vartheta}}$


Substituting these into the original function:

$\int{\frac{\frac{4}{9}cos\vartheta}{4sin\vartheta} \times \frac{\frac{-2}{9}sin\vartheta}{\sqrt{\frac{4}{9}cos\vartheta}}d\vartheta}$

$\int{\frac{\frac{4}{9}cos\vartheta}{4sin\vartheta} \times \frac{\frac{-2}{9}sin\vartheta}{\frac{2}{3}\sqrt{cos\vartheta}}d\vartheta}$

$\int{\frac{1}{9} \times \frac{cos\vartheta}{sin\vartheta} \times \frac{-1}{3} \times \frac{sin\vartheta}{\sqrt{cos\vartheta}}d\vartheta}$

$\frac{-1}{27}\int{\frac{cos\vartheta}{\sqrt{cos\vartheta}}d\vartheta}$

At this point, I got stuck and didn't know what to do. I tried using a modified version of the identity above and just went on to see if I could get anywhere but failed.

Thanks

 

[prosteve037]

Oh wow. Okay. Yeah I know what I did wrong now. Thanks everyone!

I'm still a little confused about choosing which leg to be the radical though. Does it matter? I was taught to just put the radical term on the opposite side of the reference angle. But are there instances in which this won't work?

 

[GreenPrint]

just think about it
c^2 = a^+b^2

in order to get something in the form sqrt(z^2 - y^2)
z can only equal one thing, the hypotenuse and y can be any leg

 

[SammyS]

Oh wow. Okay. Yeah I know what I did wrong now. Thanks everyone!

I'm still a little confused about choosing which leg to be the radical though. Does it matter? I was taught to just put the radical term on the opposite side of the reference angle. But are there instances in which this won't work?

No, it doesn't matter.