Question

How do I integrate this.

Answer 1

\[\int\limits_{}^{}\frac{ x }{ x^2 + 1} dx\]

Answer 2

Oo nice easy substitution for this one :)

Answer 3

\[\large\rm \int\limits\frac{x~dx}{x^2+1}\]
\[\large\rm u=x^2+1\]Differentiating should give you something close to your numerator for your du.

Answer 4

\[\large\rm du=?\]

Answer 5

2x dx

Answer 6

\[\large\rm du=2\color{orangered}{x~dx}\]Ok great.
And the numerator of our integral is this orange part, ya?
Hmm anyway we can isolate that? :)

Answer 7

\[\frac{ 1 }{ 2 }du = x dx\]

Answer 8

\[\large\rm \color{orangered}{\frac{1}{2}du=x~dx}\]Oo nice :)

Answer 9

Understand how to plug in the pieces of our substitution?\[\large\rm \int\limits\limits\frac{\color{orangered}{x~dx}}{x^2+1}=?\]

Answer 10

\[\frac{ 1 }{ 2 } \int\limits \frac{ 2x }{ x^2+1 }\]

Answer 11

?

Answer 12

You could apply that trick, introducing a 1/2 and a 2.
That would be useful if we had left it as du=2x dx.

But since we were able to turn it into x dx, we can just replace the orange with the orange,\[\large\rm \int\limits\limits\limits\frac{\color{orangered}{x~dx}}{x^2+1}=\int\limits\limits\limits\frac{\color{orangered}{\frac{1}{2}du}}{x^2+1}\]

Answer 13

And then of course our denominator is what we were calling u, yes?\[\large\rm \int\limits\limits\limits\limits\frac{\color{orangered}{x~dx}}{x^2+1}=\int\limits\limits\limits\limits\frac{\color{orangered}{\frac{1}{2}du}}{x^2+1}=\frac{1}{2}\int\limits\limits\frac{du}{u}\]

Answer 14

What do you think? Confusing? :o

Answer 15

Not yet. xP

Answer 16

And from there, this is one of your "special" integrals :)
No power rule for this one.\[\large\rm \frac{1}{2}\int\limits \frac{1}{u}du\]Remember this one?

Answer 17

ln |u|?

Answer 18

yes! but what happened to the 1/2?

Answer 19

Looks good \c:/
with a constant of integration on the end ya?

Answer 20

1/2 ln|u| + C

Answer 21

niceeee, let's undo the substitution, ya? :)

Answer 22

Cool. Appreciate your sharing your work!

Answer 23

But what's the valule of u? Re-susbstitute. u = ?

Answer 24

1/2 ln (x^2 + 1) + C

Answer 25

yay good job \c:/

Answer 26

Except that the 1/2 should be enclosed in parentheses, great job!

Answer 27

Can I try another question? On my own and you can see if it's right?

Answer 28

sure

Answer 29

Always. If you have another qu., however, would you please post it separately from this post? Thanks.

Answer 30

Okay. I'll move it and tag.