Integral : (x^2)/((x^2+1)^2)

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The easiest way to solve this integral is by using integration by parts, which is the method used in the original answer. You can also use a trigonometric substitution, as shown in Bhorok's answer, but that method can be more complicated. In summary, the answer to the integral of (x^2)/((x^2+1)^2) is (1/2)(arctan(x)-(x/x^2+1)) and the easiest way to solve it is by using integration by parts.
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The answer of the integral of (x^2)/((x^2+1)^2) is (1/2)(arctan(x)-(x/x^2+1))

In class, we've seen the steps to solve this integral, but I don't understand certain steps..
Can someone explain me how to solve this integral, step by step?
 
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  • #2
If you can post the steps and point out those that you didn't understand then I'm sure someone can help you.

BTW. The easiest way to do that one is "integration by parts". Have you learned this technique yet?
 
  • #3
uart said:
If you can post the steps and point out those that you didn't understand then I'm sure someone can help you.

BTW. The easiest way to do that one is "integration by parts". Have you learned this technique yet?

Sure, no problem, here are the steps:

Integral((x^2)/((x^2+1)^2)dx)
= (1/2)*Integral(x d(1/(x^2+1))
= (1/2)*(x/(x^2+1))-(1/2)*Integral(1/(x^2+1)dx)
= (1/2)*(x/(x^2+1))-(arctan(x))/2

Last step is the answer

(The answer I said earlier was wrong, this is the correct answer:(1/2)*(x/(x^2+1))-(arctan(x))/2)

Thank you
 
  • #4
[tex]\int\frac{x^2}{(x^2 + 1)^2} = \int x \frac{x}{(x^2 + 1)^2}[/tex]

Using ∫u v' = uv - ∫v u',
let u = x and v' = x/(x2 + 1)2
 
  • #5
It's basically separating it into parts ie.[itex]\int \frac{x^2}{(x^2+1)^2}\rightarrow \int \frac{x}{1}.\frac{x}{(x^2+1)^2}\equiv x(x. \sin(\arctan(x)))[/itex]

as

[itex]\frac{x}{1}=\frac{1}{2}x^2[/itex]

and [itex]x\frac{x}{(1+x)^2}=x.\sin(\arctan(x))[/itex]

By the trig identity.

Thus the answer is:

[itex]\int\frac{x^2}{(x^2+1)^2}=-\frac{1}{2}.\frac{x}{(x^2+1)}+\frac{1}{2}\arctan(x)+C[/itex]

Don't forget the constant of integration, it's a silly way to loose marks. :smile:
 
Last edited:
  • #6
Thank you both!
 
  • #7
Alexx1 said:
Thank you both!

np Bhorok's answer is more elegant and easier, but I thought you might need a long winded explanation and there's often more than one way to swing a cat I guess. Hope it helped. :smile:
 
  • #8
Alexx1 said:
The answer of the integral of (x^2)/((x^2+1)^2) is (1/2)(arctan(x)-(x/x^2+1))

In class, we've seen the steps to solve this integral, but I don't understand certain steps..
Can someone explain me how to solve this integral, step by step?

since you have the answer, take its derivative & work backwards. that's how to figure it out. just don't show anyone your rough work :tongue2:
 
  • #9
Alexx1 said:
Sure, no problem, here are the steps:

Integral((x^2)/((x^2+1)^2)dx)
= (1/2)*Integral(x d(-1/(x^2+1))
= (-1/2)*(x/(x^2+1)) -(1/2)*Integral(1/(x^2+1)dx)
= -(1/2)*(x/(x^2+1))+(arctan(x))/2

Last step is the answer

(The answer I said earlier was wrong, this is the correct answer:(1/2)*(x/(x^2+1))-(arctan(x))/2)

Thank you

No the original answer was correct, you dropped a minus sign in the first line of this derivation.
 

1. What is the integral of (x^2)/((x^2+1)^2)?

The integral of (x^2)/((x^2+1)^2) is equal to 1/2arctan(x) + C, where C is the constant of integration.

2. How do you solve the integral of (x^2)/((x^2+1)^2)?

To solve the integral of (x^2)/((x^2+1)^2), you can use the substitution method by letting u = x^2 + 1. This will result in the integral becoming 1/(2u^2) du, which can then be solved using the power rule.

3. Can the integral of (x^2)/((x^2+1)^2) be simplified further?

No, the integral of (x^2)/((x^2+1)^2) cannot be simplified further. It is already in its most simplified form.

4. What is the domain of the integral of (x^2)/((x^2+1)^2)?

The domain of the integral of (x^2)/((x^2+1)^2) is all real numbers.

5. Is the integral of (x^2)/((x^2+1)^2) an even or odd function?

The integral of (x^2)/((x^2+1)^2) is an odd function, as it satisfies the property f(-x) = -f(x).

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