- #1

Alexx1

- 86

- 0

**(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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- Thread starter Alexx1
- Start date

- #1

Alexx1

- 86

- 0

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?

- #2

uart

Science Advisor

- 2,797

- 21

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

- #3

Alexx1

- 86

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BTW. The easiest way to do that one is "integration by parts". Have you learnt 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:

Thank you

- #4

Bohrok

- 867

- 0

Using ∫u v' = uv - ∫v u',

let u = x and v' = x/(x

- #5

Dunebug7

- 4

- 0

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.

[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.

Last edited:

- #6

Alexx1

- 86

- 0

Thank you both!

- #7

Dunebug7

- 4

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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.

- #8

fourier jr

- 757

- 13

(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

uart

Science Advisor

- 2,797

- 21

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.

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