Improper Integral with trig integral

[shubox]

Homework Statement

∫(dx/((1+x^2)^2) from 0 to ∞
Determine whether the improper integral converges and if so, evaluate it.

Homework Equations

1+ tan^2(x) = sec^2(x)
1/sec(x) = cos(x)

The Attempt at a Solution

Initially I had no idea how to approach this problem. The answer in the back of the book is ∏/4, which tells me that maybe trig integrals are involved. So i started off with:
lim(R→∞) ∫(dx/((1+x^2)^2) from 0 to R.
x=tan(x)
lim(R→∞) ∫(dx/((1+tan^2(x))^2) from 0 to R.
=lim(R→∞) ∫(dx/(sec^4(x)) from 0 to R.
I do not know where to go from here. Any help would be appreciated

 

[SammyS]

Homework Statement

∫(dx/((1+x^2)^2) from 0 to ∞
Determine whether the improper integral converges and if so, evaluate it.

Homework Equations

1+ tan^2(x) = sec^2(x)
1/sec(x) = cos(x)

The Attempt at a Solution

Initially I had no idea how to approach this problem. The answer in the back of the book is ∏/4, which tells me that maybe trig integrals are involved. So i started off with:
lim(R→∞) ∫(dx/((1+x^2)^2) from 0 to R.
x=tan(x)
lim(R→∞) ∫(dx/((1+tan^2(x))^2) from 0 to R.
=lim(R→∞) ∫(dx/(sec^4(x)) from 0 to R.
I do not know where to go from here. Any help would be appreciated

It's not a good idea to use the same variable in your substitution.

Let x=tan(θ) then dx = sec²(θ) dθ .

After finding the anti-derivative in terms of θ, change back to x, plug in the limits of integration, then take the limit.

 

[shubox]

Oh I see now. Subsituting in dx cancels out sec²(θ) which then results in θ. Replacing θ with x gives arc tangent and then solving the integral from -∞ to ∞ results in ∏.
Thanks for helping

 

[SammyS]

Oh I see now. Subsituting in dx cancels out sec²(θ) which then results in θ. Replacing θ with x gives arc tangent and then solving the integral from -∞ to ∞ results in ∏.
Thanks for helping

Not quite.

The denominator is (sec²(θ))² .

Then use cos²(θ) = (1/2)(cos(2θ) + 1) .

 

[HallsofIvy]

By the way, $tan(\theta)$ goes to infinity as $\theta$ goes to $\pi/2$. So once you have made the substitution, instead of going to infinity, you limit should be going to $\pi/2$.

 

[shubox]

Ah I forgot the quantity squared. But after doing that I got the same thing: 1/2(cos(2θ)+1)
Taking the integral of this from ∏/2 to -∏/2 results in ∏/2 which is the answer in the back of the book.

sweeet, makes sense now. thanks a lot.