Another (quick) Trig substitution question

[ChaoticLlama]

for the integral

∫√(1 + 4x²) / x^4

I know that I make the substitution x = tanθ

but how do I find the solution for the radical?

usually radicals are in the form of

√(a² + x²)

let x = tanθ

√(a² + a²tan²θ)

√(a²(1 + tan²θ))

√(a²sec²θ)

asecθ

What do I do in this case?

 

[tkhunny]

for the integral
∫√(1 + 4x²) / x^4
I know that I make the substitution x = tanθ

What? Maybe x = (1/2)*tan(u) and dx = (1/2)*(1+tan²(u))*du

However, will that get you anywhere? I think you're barking up the wrong tree.

Try integration by parts.
u = sqrt(1+4*x^2)
dv = x^(-4)*dx

THEN maybe you can try your trig substitution.

 

[soroban]

Hello, ChaoticLlama!

For the integral: ∫√(1 + 4x²) / x^4

I know that I make the substitution x = tanθ . . . no

Use: . 2x .= .tanθ . . . then: . 2 dx .= .sec²θ dθ
. . . . . . ______ . . . . . _______ . . . . . ____
And: . √1 + 4x² . =. √1 + tan²θ . = . √sec²θ . = . secθ

Now make the substitution . . .