Question

Help!!!
Integrate ln | 1+x |

Answer 1

\[\Large \int \ln|1+x|dx\]
this?

Answer 2

yeah

Answer 3

Well then, all you need for this is to recall the way to integrate *by parts*
and the derivative of the natural logarithm.

INTEGRATION BY PARTS:\[\Large \int udv = uv - \int vdu\]
\[\Large \frac{d}{dt}\ln|t| = \frac1t\]

Answer 4

how to integrate by parts if there is only one part?

Answer 5

Good question... much like how the skin is so... second nature, people don't usually call it an organ like, say, the liver, the stomach, etc

But I digress ^_^

THESE are two parts in themselves... people hardly seem to count the differential dx as a part in itself :P
\[\Large \int \color{green}{\ln|1+x|}\color{red}{dx}\]

Answer 6

So let's begin.
Let u = ln|1+x|
and dv = dx

Can you find du and v?

Answer 7

sure

Answer 8

du= 1/1+x
v=x

Answer 9

is it?

Answer 10

That is correct :)
Apply the formula:
\[\Large = x\ln|1+x| - \color{blue}{\int \frac{x}{1+x}dx}\]

I trust that 'blue' integral won't be a problem? ^_^

Answer 11

for da blue part do integration by parts too??

Answer 12

Nope.
\[\Large \frac{x}{1+x}= \frac{x\color{red}{+1-1}}{1+x}= \frac{x+1}{x+1}-\frac1{1+x}=1-\frac1{1+x}\]
\[\Large \implies \int\frac{x}{1+x}dx=\int\left[1-\frac1{1+x}\right]dx\]
Much easier to integrate, no? ^_^

Answer 13

Much of the time, integration calls for creativity, so keep this technique in mind, and use it if the situation calls for it ^_^

Answer 14

how can we just put +1 n -1 like tat?

Answer 15

We can... we basically added zero ^_^
+1-1 = 0
So it does nothing, except give us an *equivalent* expression which just happened to be easier to integrate.

Such tricks come a plenty in maths, so watch out for them.

Look, I need to go now, I'm sure someone can pick up where we left off...

Good luck ^_^

-----------------------------------
Terence out
:)

Answer 16

:'(
Thanks btw
i'll try my best