anilorap
i want to confirm the answer of... integrate lnx/x dx
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anilorap
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my answer is ln lxl +c
myininaya
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let u=lnx
du=dx/x
\[\int\limits_{}^{}u du=\frac{u^2}{2}+C=\frac{(lnx)^2}{2}+C\]
we can even check this:
\[(\frac{(lnx)^2}{2}+C)'=2*\frac{lnx}{2}*\frac{1}{x}+0=\frac{lnx}{x}\]
myininaya
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do you got it?
anilorap
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yes
myininaya
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k :)
anilorap
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q. how do i know what to choose?
myininaya
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i was looking for \[\int\limits_{}^{}f(x)*f'(x)dx\]
myininaya
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sometimes its not always that easy though
myininaya
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someone said it is like an art and i agree
anilorap
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jajaj..ok
myininaya
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it is not always easy to know what substititon to make but since it was in the form of above it was easy
let u=f(x)
du=f'(x) dx
myininaya
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\[\int\limits_{}^{}\frac{lnx}{x}dx=\int\limits_{}^{}lnx*\frac{1}{x} dx\]
myininaya
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so it was easy for me to notice pretty fast that (lnx)'=1/x
so thats why i let u=lnx
anilorap
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ok..perfect
anilorap
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so how can i integrate tanx dx..... even though i know thats it is equal to -lnlcosxl+c
myininaya
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\[\int\limits_{}^{}\frac{sinx}{cosx} dx\]
right?
myininaya
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tanx=sinx/cosx agree?
anilorap
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ohhhhhhhh... ok
anilorap
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yes
anilorap
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so that du/u
myininaya
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now what do you think the substitution will be?
anilorap
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with the negative sign of the sin
anilorap
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u=cos
anilorap
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du=sin x
anilorap
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sorry
du=-sinx
myininaya
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du=-sinx dx (i like not to forget about this dx part)
anilorap
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oh yea
myininaya
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so show me what that gives us don't integrate yet
anilorap
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then u have
integral of du/u
myininaya
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negative or positive?
anilorap
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- integral of du/u
myininaya
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yes and now integrate
anilorap
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so my result will be
-ln/u/+c
anilorap
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-ln/cosx/+c
myininaya
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yes :)
anilorap
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yeaaa
thank uuuu
myininaya
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np
myininaya
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so whats this \[\int\limits_{}^{}-tanx dx\] equal to?
anilorap
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ln/cosx/+c
anilorap
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becuase the negative sign will change to postive once u take the derivative of cosx
anilorap
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why u pic u =sinx?
myininaya
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accident
myininaya
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u=cosx
du=-sinx dx
anilorap
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:)
anilorap
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good....:
myininaya
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you want to try one i made up? its not hard
anilorap
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ok
myininaya
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\[\int\limits_{}^{}\frac{x+1}{x^2+2x}dx\]
anilorap
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ok.. let me try
anilorap
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u=x^2 + 2x
du=2x+2 dx
=2(x+1) dx
so,
1/2du=x+1 dx
myininaya
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very good so far
anilorap
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1/2 integral of du/u
=1/2 lnlul+c
=1/2 ln/x^2+2xl+c
myininaya
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:)
anilorap
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yeaaaaaaaaaaaaa
anilorap
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i love this stuff
myininaya
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lol it is nice
myininaya
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\[\int\limits_{}^{}\frac{f'(x)}{f(x)}dx\]
so when we have this what will out answer be in the form of?
anilorap
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f(x)*f'(x)
myininaya
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look at the one you just did
didn't we have f'/f dx?
anilorap
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ohh... sorry.. yes yes
myininaya
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=ln|f(x)|+C
anilorap
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yes
myininaya
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\[\int\limits_{}^{}[f(x)]^n*f'(x)dx\]
\[n \neq -1\]
what about this?
what will our answer be?
myininaya
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i will give you a hint
let u=f(x)
so du=f'(x) dx
anilorap
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jjajjjaj... nice hint
myininaya
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where are you from?
you say jajaja alot lol
anilorap
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dominican republic
anilorap
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i am just laughing
myininaya
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jajaaja= laughing?
anilorap
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yess... how can i laugh??
myininaya
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its fine the way you type your laughter
no worries
i like it
anilorap
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lol
myininaya
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so this all makes a little more sense?
myininaya
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the integral stuff?
anilorap
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why?
myininaya
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im asking do you get the integral stuff better?
anilorap
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ohhhhhh,, yess
myininaya
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good :) im fixing to go to sleep goodnight
anilorap
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for me is mornig.. but good nigth,,,
myininaya
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good morning*
i didn't sleep all night lol
anilorap
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jajajajaaja
anilorap
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ayayay.. no good
anilorap
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thank u very much...
myininaya
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np
if you want to try another
this is a good one
\[\int\limits_{}^{}x*\sqrt{x+1} dx\]
the answer is given above in one of the other threads if you want to check yourself
myininaya
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later
anilorap
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ok,, thanks