How can I correctly solve the integral of (x^2)/((x^2+1)^2) step by step?

[Alexx1]

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?

 

[uart]

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?

 

[Alexx1]

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

 

[Bohrok]

$$\int\frac{x^2}{(x^2 + 1)^2} = \int x \frac{x}{(x^2 + 1)^2}$$

Using ∫u v' = uv - ∫v u',
let u = x and v' = x/(x² + 1)²

 

[Dunebug7]

It's basically separating it into parts ie.$\int \frac{x^2}{(x^2+1)^2}\rightarrow \int \frac{x}{1}.\frac{x}{(x^2+1)^2}\equiv x(x. \sin(\arctan(x)))$

as

$\frac{x}{1}=\frac{1}{2}x^2$

and $x\frac{x}{(1+x)^2}=x.\sin(\arctan(x))$

By the trig identity.

Thus the answer is:

$\int\frac{x^2}{(x^2+1)^2}=-\frac{1}{2}.\frac{x}{(x^2+1)}+\frac{1}{2}\arctan(x)+C$

Don't forget the constant of integration, it's a silly way to loose marks.

 

[Alexx1]

Thank you both!

 

[Dunebug7]

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.

 

[fourier jr]

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:

 

[uart]

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.