Antiderivative of the following function

[KataKoniK]

How would you guys come about getting the antiderivative for the following function?

8u^5 / ((4 + u^4)^2)

I tried rearranging the equation like the following:

8u^5 * ((4 + u^4)^(-2)), but it's not really getting me anywhere. I tried to do a u substitution, but can't find one where it knocks off the top u^5.

Thanks in advance.

 

[SteveDB]

I take it that you're taking the A.D. wrt u?
My TI89 gives:
pi*atan(u^2/2)/180 - (2*u^2/(u^4+4))
Which in turn tells me that your calculus text should have a table for it in one of the appendices.
Look for it in the TOC, or index.
Look for an indefinite integral that has the general form:
a*x^(n)/(x^(n-1) +(n-1))^(2)
Or something similar.

 

[Jameson]

Integration by parts and you'll have to use trig substitution. That's a nasty one. Good luck!

 

[hotvette]

Here are a couple of hints. Try the substitution x = u². That should help reduce things to a more manageable integrand. After that, a little clever algebra and a trig substitution will get you there. Not a trivial problem.

hotvette

 

[KataKoniK]

Here are a couple of hints. Try the substitution x = u². That should help reduce things to a more manageable integrand. After that, a little clever algebra and a trig substitution will get you there. Not a trivial problem.

hotvette

Hmm, any hint on what the clever algebra should be? I can't really figure it out from here. I have simplified the equation down to

4x² / (4 + x²)² dx

using x = u²

 

[hotvette]

Try adding and subtracting the same number from the top. Gather terms and split the problem into 2 problems.

Example: if I have y in the numerator, y - 2 + 2, or (y - 2) + 2 is the same thing. Just pick the right number to add and subtract.

 

[KataKoniK]

Thanks for the reply. Am I on the right track?


4x² / (4 + x²)² dx
= (4x² + 16 - 16) / (4 + x²)² dx
= 4((x² + 4) - 4)) / (4 + x²)² dx
= (4 / (4 + x²)) - (4/ (4 + x²)²) dx

 

[hotvette]

You bet. The first one is readily integrated and the second is w/ a trig substitution. My college calc book has an entire section on appropriate trig substitutions based on what the integrand looks like - that's where I found the appropriate trig substituion.

 

[KataKoniK]

Thanks. However, can you spot my mistake when solving for the antiderivative of (4/ (4 + x²)²)? I must be doing something wrong, but I can't find what

(4/ (4 + x²)²)

Let x = 2 tan theta
dx = 2 sec²theta dtheta

= (8 sec²theta / ((4 + 4tan²theta)²)

factor out a 2

= 4 + 4 tan² theta / ((4 + 4tan²theta)²)
= 1 / (4 + 4tan²theta)
= tan inverse (tan theta)

 

[hotvette]

The part I don't follow is how you get from:

$$\frac {d \Theta}{(4 + 4 tan^2 \Theta)}$$

to

$$tan^{-1} tan \Theta$$

By the way, I had a slightly different integrand, with 16 in the numerator instead of 4, but frankly, I don't want to take the time to re-check it. Mine might be wrong, I don't know. My objective here is getting you going on the right path, not necessarily verifying the exact answer.

 

[KataKoniK]

Alright, thanks. I'll just try it again.

 

[Jameson]

Hmm, any hint on what the clever algebra should be? I can't really figure it out from here. I have simplified the equation down to

4x² / (4 + x²)² dx

using x = u²

I think there's a problem in this substitution.

$$\int \frac{8u^5}{(4+u^4)^2}du$$

$$u=x^2$$
$$du=2xdx$$

 

[hotvette]

I think there's a problem in this substitution.

$$\int \frac{8u^5}{(4+u^4)^2}du$$

$$u=x^2$$
$$du=2xdx$$

No. The substitution is $x=u^2$. You have it backwards.

$$\int \frac{4x^2}{(4+x^2)^2} \ dx$$

is correct after the substitution.

 

[hotvette]

I just checked my work again. Here's what I get:

$$\int \frac{4x^2}{(4+x^2)^2} \ dx \ = \ \int \frac{4}{(4 + x^2)} \ dx \ - \int \frac{16}{(4 + x^2)^2} \ dx$$

Btw, it might be prudent to reduce even further by factoring out the 4's to make it (1 + something^2) instead of (4 + x^2). May not be critical, though.

 

[KataKoniK]

Yup, that's what I got after the substitution and working. Sorry for replying so late. I finally had a chance to do this question just now.


$$\int \frac{16}{(4 + x^2)^2} \ dx = \frac{32sec^{2}\Theta}{(4 + 4tan^{2}\Theta)^2}$$ d theta

where $$x = 2tan\Theta$$
$$\ dx = 2 sec^{2}\Theta$$ d theta

After all that, the result I get is $$\Theta + \frac{sin2\Theta}{2}$$, but for some reason, when I use my calculator to see if this answer is correct, it doesn't match up with the derivative values of the original equation 8u^5 / (4 + u^4)^2

 

[hotvette]

Don't forget the first integral.

 

[KataKoniK]

Yup, I checked it with both the first and the second integral and no dice. The antiderivative for

$$\int \frac{4}{(4 + x^2)} \ dx$$

is $$2tan^{-1}(x/2)$$