Int 1

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January 16th 2013, 06:05 AM
Boo

Int 1

All!
Please, can someone help me with:

$\int\frac{x-3}{x^2+1}dx$

The result should be:
$\frac{1}{2}ln|x^2+1|-3arctanx+C$ (Worried)

Many thanks!
January 16th 2013, 06:17 AM
sbhatnagar

Re: Int 1

Quote:

Originally Posted by Boo

All!
Please, can someone help me with:

$\int\frac{x-3}{x^2+1}dx$

The result should be:
$\frac{1}{2}ln|x^2+1|-3arctanx+C$ (Worried)

Many thanks!

Write this as

$\int\frac{x}{x^2+1}dx-3\int \frac{1}{1+x^2}dx$

For the first integral, substitute $u=1+x^2$ .

$\int \frac{x}{x^2+1}dx = \frac{1}{2}\int \frac{1}{u}du$

$=\frac{1}{2}\ln |u|+\text{Constant} = \frac{1}{2}\ln |1+x^2| +\text{Constant}$

For the second integral, substitute $x=\tan \theta$ .

$\int \frac{1}{1+x^2}dx = \int \frac{\sec^2 \theta}{1+\tan^2 \theta}d\theta$

Note that $1+\tan^2 \theta =\sec^2 \theta$ , so our integral will become

$\int d\theta = \theta + \text{Constant} = \arctan(x)+\text{Constant}$

Combine these two results to obtain

$\int\frac{x}{x^2+1}dx-3\int \frac{1}{1+x^2}dx$

$= \frac{1}{2}\ln|1+x^2|-3\arctan(x)+\text{Constant}$
January 19th 2013, 08:22 AM
Boo

Re: Int 1

Many thanks, sb!!!
please, how did U get the second line? What is the rule?
Many thanks!!!
January 19th 2013, 11:57 AM
HallsofIvy

Re: Int 1

Have you never taken a Calculus class? What sbhatnagar did is pretty standard. To integrate $\int \frac{x}{x^2+ 1}dx$ , seeing that " $x^2$ " in the denominator and the "xdx" in the numerator, you should immediately think to try the substitution $u= x^2+ 1$ so that $du= 2xdx$ or $\frac{1}{2}du= x dx$ . We can write
$\int \frac{x dx}{x^2+ 1}= \int\frac{\frac{1}{2}du}{u}= \frac{1}{2}\int \frac{du}{u}$ .

And now you should recognise that $\frac{d ln(u)}{du}= \frac{1}{u}$ so that $\int \frac{du}{u}= ln|u|+ C$ .
(Note the absolute value: ln|u|. ln(u) is only defined for postive u and u, here, might be negative.)
January 20th 2013, 06:07 AM
Boo

Re: Int 1

Thank You , Hallsofivy, that was really helpful!!!AM really new here!
January 20th 2013, 06:10 AM
Boo

Re: Int 1

Please, how did he get this with x_=tan fi??
many thanks!
p.s. trying to google it but it is not that easy...if someone has good pleages please send it...am new here...MANY THANKS!!!
January 20th 2013, 12:16 PM
tom@ballooncalculus

Re: Int 1

Just in case a picture helps...

http://www.ballooncalculus.org/draw/...l/fourteen.png

That's how the substitution works. However, you can see that it depends on the chain rule, and I would prefer to see how we are working backwards through a differentiation according to the chain rule.

If, like me, you were taught to see differentiation as a downwards journey and integration up, then you'll see the logic of this...

http://www.ballooncalculus.org/draw/.../fourteena.png

... where (key in spoiler) ...

Spoiler:

http://www.ballooncalculus.org/asy/chain.png

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

The general drift is...

http://www.ballooncalculus.org/asy/diffChain.png

So, the bottom row shows that we have multiplied the derivative with respect to the inner function by the derivative of the inner function itself. Which is what the chain rule wants.

However, the inner function is just x, so the derivative of that is just one, so no big deal from applying the chain rule.

But now, carefully, remove the inner function (by twisting gently) and replace with tan theta...

http://www.ballooncalculus.org/draw/.../fourteenb.png

Now the derivative of the inner function is sec squared so, as sbhatnagar explains, we are really integrating (or anti-differentiating) one, with respect to theta...

http://www.ballooncalculus.org/draw/.../fourteenc.png

Then just replace the original inner function...

http://www.ballooncalculus.org/draw/.../fourteend.png

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!
January 20th 2013, 12:52 PM
tom@ballooncalculus

Re: Int 1

The first integral is of the other main kind (that involve a substitution). It comes ready with an appropriate inner function...

http://www.ballooncalculus.org/draw/.../fourteene.png

So you only need to deduce that the function F is ln...

http://www.ballooncalculus.org/draw/.../fourteenf.png

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!