Calculating 4 Integral: A Tan & Cos Theta Substitution

[flybyme]

Homework Statement

how would one calculate 4 \int_0^{\frac{\pi}{2}} \frac{\cos^2 \theta}{(1 + \cos^2 \theta)^2} d \theta?

The Attempt at a Solution

someone suggested a u = \tan \theta substitution, but i don't understand why and how this would help me. couldn't i just use u = \cos t?

 

[AlephZero]

Divide top and bottom by \cos^4\theta.

The function becomes \frac{\sec^2\theta}{(\sec^2\theta+1)^2}

Does that make u=\tan\theta looks like a good move?

u = cos t is probably a bad move, because du = -sin t dt and the sin t doesn't cancel with anything.

 

[flybyme]

Divide top and bottom by \cos^4\theta.

The function becomes \frac{\sec^2\theta}{(\sec^2\theta+1)^2}

Does that make u=\tan\theta looks like a good move?

the only trig functions we learn here are tan, cos and sin, so I'm not really sure... :)

u = cos t is probably a bad move, because du = -sin t dt and the sin t doesn't cancel with anything.

wouldn't i get du = -sin t dt = -sqrt(1 - cos^2 t) = -sqrt(1 - u^2) dt?

because then the integral should translate to -4 \int_1^0 \frac{u^2}{ (u^2+1)(1-u^2)^{1/2} } du, or am i wrong?

 

[drpizza]

the only trig functions we learn here are tan, cos and sin, so I'm not really sure... :)

I agree with what you had in the last post, except, wouldn't it be (1+u^2)^2 in the denominator? Anyway, now what?

 

[flybyme]

yeah, I've never understood why that is so...

 

[mjsd]

the only trig functions we learn here are tan, cos and sin, so I'm not really sure... :)



wouldn't i get du = -sin t dt = -sqrt(1 - cos^2 t) = -sqrt(1 - u^2) dt?

because then the integral should translate to -4 \int_1^0 \frac{u^2}{ (u^2+1)(1-u^2)^{1/2} } du, or am i wrong?

what is the derivative of \tan \theta?

 

[mjsd]

also do you know 1+\tan^2\theta =\sec^2\theta?

 

[drpizza]

I edited my post just a little too late... isn't it (u^2 + 1)^2 in the denominator?

 

[flybyme]

I edited my post just a little too late... isn't it (u^2 + 1)^2 in the denominator?

oops. yeah, that's right.

 

[flybyme]

also do you know 1+\tan^2\theta =\sec^2\theta?

no, i didn't know that. :)

so i would get 4 \int_0^{\pi/2} \frac{\sec^2 \theta}{(\sec^2 \theta + 1)^2} d \theta = 4 \int_0^{\pi/2} \frac{\tan^2 \theta + 1}{(\tan^2 \theta + 2)^2} d \theta = 4 \int_0^{\Omega} \frac{u^2 + 1}{(u^2 + 2)^2(u^2 + 1)} du = 4 \int_0^{\Omega} \frac{du}{(u^2 + 2)^2}

as u = \tan \theta and du = (1 + \tan^2 \theta) d \theta = (1 + u^2) d \theta.

does that seem correct? I'm unsure on what \Omega would be though, as \tan \pi/2 = \infty

 

[Gib Z]

Ahh Nope, \int \frac{\sec^2\theta}{(\sec^2\theta+1)^2} d\theta becomes \int \frac{1}{u^2} du when u = tan theta, because the derivative of tan theta is (sec theta)^2.

In case you don't know Sec x = 1/(cos x) by definition.

 

[AlephZero]

Ahh Nope, \int \frac{\sec^2\theta}{(\sec^2\theta+1)^2} d\theta becomes \int \frac{1}{u^2} du when u = tan theta

Oops... you mean \int \frac{1}{(u^2+2)^2} du

1 + tan^2 x = sec^2 x

NOT 1 + sec^x = tan^2 x !

(And even if that was true, it would have been u^4 not u^2)

 

[Gib Z]

Sorry about that, your right of course AlephZero.

 

[dextercioby]

\int_{0}^{\frac{\pi }{2}}\frac{\cos ^{2}x}{\left( 1+\cos ^{2}x\right) ^{2}}dx=\allowbreak \int_{0}^{\infty }\frac{1}{\left( 1+p^{2}\right) ^{2}\left( 1+\frac{1}{1+p^{2}}\right) ^{2}}\,dp=\int_{0}^{\infty }\frac{1}{\left( 2+p^{2}\right) ^{2}}\,dp\allowbreak

=\allowbreak \int_{0}^{\frac{1}{2}\pi }\frac{\sqrt{2}}{4+4\tan ^{2}q}\,dq=\frac{\sqrt{2}}{4}\int_{0}^{\frac{1}{2}\pi }\cos ^{2}q \ {} \ dq=\frac{\sqrt{2}}{4}\frac{\pi }{4}=\frac{\pi\sqrt{2}}{16}

 

[Gib Z]

Would I be the only person to be a tiny bit confused as to what exactly dexter did ...

 

[dextercioby]

Would it help if you knew that \tan x=p and p=\sqrt{2}\tan q ?

 

[Gib Z]

Ahh yup it did :D