Calculus - Integration

**bhuang** · January 9th 2010, 08:33 PM

how would I go about integrating:

[x^2 / (1+x^2)] dx

**11rdc11** · January 9th 2010, 08:39 PM

Originally Posted by bhuang

how would I go about integrating:

[x^2 / (1+x^2)] dx

use long division and get

$\int \bigg(1 - \frac{1}{1+x^2}\bigg)dx$

Then you should remember the second part is arctan

**Prove It** · January 9th 2010, 08:45 PM

Originally Posted by 11rdc11

use long division and get

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

Then you should remember the second part is arctan

And if you didn't know that, you could use trigonometric substitution.

$\int{1 - \frac{1}{1 + x^2}\,dx} = \int{1\,dx} - \int{\frac{1}{1 + x^2}\,dx}$

$= x - \int{\frac{1}{1 + x^2}\,dx}$ .

To find $\int{\frac{1}{1 + x^2}\,dx}$

Let $x = \tan{\theta}$ so that $dx = \sec^2{\theta}\,d\theta$ .

Also note that $\theta = \arctan{x}$ .

So the integral becomes

$\int{\frac{1}{1 + \tan^2{\theta}}\,\sec^2{\theta}\,d\theta}$

$= \int{\frac{1}{\sec^2{\theta}}\,\sec^2{\theta}\,d\t heta}$

$= \int{1\,d\theta}$

$= \theta + C$

$= \arctan{x} + C$ .

So $\int{1 - \frac{1}{1 + x^2}\,dx} = x - \arctan{x} + C$ .

**bhuang** · January 9th 2010, 09:04 PM

can you do the long division part?

**Prove It** · January 9th 2010, 09:08 PM

Originally Posted by bhuang

can you do the long division part?

Of course you can.

$\frac{x^2}{1 + x^2} = \frac{1 + x^2 - 1}{1 + x^2} = \frac{1 + x^2}{1 + x^2} - \frac{1}{1 + x^2} = 1 - \frac{1}{1 + x^2}$ .

**bhuang** · January 9th 2010, 09:14 PM

i haven't learned arctan or trigonometry for integration. so can I not simply integrate 1/(1+x^2) as ln(1+x^2)?
so my final answer would be
x-ln(1+x^2)+c

**11rdc11** · January 9th 2010, 09:16 PM

Originally Posted by bhuang

i haven't learned arctan or trigonometry for integration. so can I not simply integrate 1/(1+x^2) as ln(1+x^2)?
so my final answer would be
x-ln(1+x^2)+c

No since $u = x^2$ so $du=2xdx$ which you do not have in the numerator.

**Prove It** · January 9th 2010, 09:19 PM

Originally Posted by bhuang

i haven't learned arctan or trigonometry for integration. so can I not simply integrate 1/(1+x^2) as ln(1+x^2)?
so my final answer would be
x-ln(1+x^2)+c

It should also be clear that $\arctan{x} \neq \ln{(1 + x^2)}$ .

I showed you how to solve it using trigonometric substitution...

**bhuang** · January 9th 2010, 09:23 PM

I don't want to use something I haven't learned, it will confuse me before my test. I might not even remember it if I come across a similar question. I never made u=x^2.

**11rdc11** · January 9th 2010, 09:25 PM

Originally Posted by bhuang

I don't want to use something I haven't learned, it will confuse me before my test. I might not even remember it if I come across a similar question. I never made u=x^2.

There isn't another way to solve this but the way Prove It and I showed you. What did you use as your u then?

**bhuang** · January 9th 2010, 09:28 PM

I didn't have a u. I went from 1-1/(1+x^2) and I integrated.
1 becomes x, when integrated. And 1/(1+x^2) I made (1+x^2)^-1 so I could integrate that into ln(1+x^2).

**11rdc11** · January 9th 2010, 09:34 PM

Originally Posted by bhuang

I didn't have a u. I went from 1-1/(1+x^2) and I integrated.
1 becomes x, when integrated. And 1/(1+x^2) I made (1+x^2)^-1 so I could integrate that into ln(1+x^2).

You aren't allowed to do that in math.

**bhuang** · January 9th 2010, 09:36 PM

Why? What's wrong with doing that?

**Prove It** · January 9th 2010, 09:49 PM

Watch carefully:

$\int{\frac{1}{x}\,dx} = \ln{|x|} + C$ by definition.

We can also see that

$\int{\frac{1}{ax + b}\,dx} = \frac{1}{a}\ln{|ax + b|}$ .

This comes from a $u$ substitution.

Let $u = ax + b$ so that $\frac{du}{dx} = a$ .

So $\int{\frac{1}{ax + b}\,dx} = \frac{1}{a}\int{\frac{a}{ax + b}\,dx}$

$= \frac{1}{a}\int{\frac{1}{u}\,\frac{du}{dx}\,dx}$

$= \frac{1}{a}\int{\frac{1}{u}\,du}$

$= \frac{1}{a}\ln{|u|} + C$

$= \frac{1}{a}\ln{|ax + b|} + C$ .

BUT you are trying to find $\int{\frac{1}{1 + x^2}\,dx}$ .

We DO NOT have a formula for that yet because it is not of the form $\frac{1}{ax + b}$ . But you could TRY a $u$ substitution.

Watch what happens.

Let $u = 1 + x^2$ .

We can see that $\frac{du}{dx} = 2x$ .

Can we change $\int{\frac{1}{1 + x^2}\,dx}$ into a function that is of the form $\int{\frac{2x}{1 + x^2}\,dx}$ ? NO, we can not.

So the rule for $\ln$ DOES NOT WORK in this case.

Now see my post regarding trigonometric substitution to show you the method that DOES work.

Alternatively, let's try differentiating $\arctan{x}$ and seeing if we get $\frac{1}{1 + x^2}$ . If we do, then $\int{\frac{1}{1 + x^2}\,dx} = \arctan{x}$ .

$y = \arctan{x}$

$\tan{y} = x$

$\frac{d}{dx}(\tan{y}) = \frac{d}{dx}(x)$

$\frac{dy}{dx}\,\frac{d}{dy}(\tan{y}) = 1$

$\sec^2{y}\,\frac{dy}{dx} = 1$

$\frac{dy}{dx} = \cos^2{y}$ .

But we know that $y = \arctan{x}$ and we also know that $\cos{\theta} = \frac{1}{\sqrt{1 + \tan^2{\theta}}}$

So $\frac{dy}{dx} = \left(\frac{1}{\sqrt{1 + \tan^2{y}}}\right)^2$

$= \frac{1}{1 + \tan^2{y}}$

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

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

So $\frac{d}{dx}(\arctan{x}) = \frac{1}{1 + x^2}$ .

Therefore $\int{\frac{1}{1 + x^2}\,dx} = \arctan{x} + C$ .

No more arguments please!