integrating log (1 Viewer)

jnney · Dec 12, 2011

can someone help me, i dont know how to do this.

http://www.wolframalpha.com/input/?i=integrate+lnx/x

http://www.wolframalpha.com/input/?i=integrate+(x/(2x+1))

4025808 · Dec 12, 2011

click the show steps procedure, then it'll show you how to do so.

Shadowdude · Dec 12, 2011

$\int \frac{ln(x)}{x} dx = \int ln(x).\frac{1}{x}\,dx$

Let u = ln(x). Solve.

SpiralFlex · Dec 13, 2011

$\int \frac{\ln x}{x}\: {\mathrm{d} x}$

$\int \ln x \frac{1}{x}\: {\mathrm{d} x}$

By the substitution method,

Let $u=\ln x$

$du = \frac{1}{x} \: {\mathrm{d} x}$

Now we get our integral as,

$\int u\: {\mathrm{d} u}$

We can integrate this with ease!

$\frac{u^2}{2}+C$

Now we replace,

$\frac{(\ln x)^2}{2}+C$

$y=\int \frac{x}{2x+1}\: {\mathrm{d} x}$

$2y=\int \frac{2x}{2x+1}\: {\mathrm{d} x}$

$2y=\int \frac{2x+1-1}{2x+1}\: {\mathrm{d} x}$

$2y=\int (1-\frac{1}{2x+1})\: {\mathrm{d} x}$

$y=\frac{1}{2}\int (1-\frac{1}{2}*\frac{2}{2x+1})\: {\mathrm{d} x}$

$\therefore y=\frac{1}{2}x-\frac{1}{4}\ln (2x+1)+C$

*Note this one is errored* I will fix when I come back from school!

such_such · Dec 13, 2011

Second Question

<a href="http://www.codecogs.com/eqnedit.php?latex=\int \frac{x}{2x@plus;1} dx" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\int \frac{x}{2x+1} dx" title="\int \frac{x}{2x+1} dx" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=let u = 2x@plus;1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?let u = 2x+1" title="let u = 2x+1" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=du=2x@plus;1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?du=2x+1" title="du=2x+1" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{du}{dx}=2" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{du}{dx}=2" title="\frac{du}{dx}=2" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{du}{2}=dx" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{du}{2}=dx" title="\frac{du}{2}=dx" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=2x@plus;1=u" target="_blank"><img src="http://latex.codecogs.com/gif.latex?2x+1=u" title="2x+1=u" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=2x=u-1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?2x=u-1" title="2x=u-1" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=x=\frac{u-1}{2}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x=\frac{u-1}{2}" title="x=\frac{u-1}{2}" /></a>

Sub all of this into the original

<a href="http://www.codecogs.com/eqnedit.php?latex=\int \frac{x}{2x@plus;1}\cdot \frac{du}{2}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\int \frac{x}{2x+1}\cdot \frac{du}{2}" title="\int \frac{x}{2x+1}\cdot \frac{du}{2}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{2}\int \frac{\frac{u-1}{2}}{u} du" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{2}\int \frac{\frac{u-1}{2}}{u} du" title="\frac{1}{2}\int \frac{\frac{u-1}{2}}{u} du" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{2}\int\frac{u-1}{2u}du" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{2}\int\frac{u-1}{2u}du" title="\frac{1}{2}\int\frac{u-1}{2u}du" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{4}\int\frac{u-1}{u}du" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{4}\int\frac{u-1}{u}du" title="\frac{1}{4}\int\frac{u-1}{u}du" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{4}[1-\frac{1}{u}]" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{4}[1-\frac{1}{u}]" title="\frac{1}{4}[1-\frac{1}{u}]" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{4}[u-logu]" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{4}[u-logu]" title="\frac{1}{4}[u-logu]" /></a>

sub u back
<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{4}[2x@plus;1-log(2x@plus;1))" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{4}[2x+1-log(2x+1))" title="\frac{1}{4}[2x+1-log(2x+1))+C" /></a>

Which is the same answer as wolframalpha, just rearranged
*And plus, there's +C at the end, but for some reason, I can't change it now*

SpiralFlex · Dec 13, 2011

such_such said:
Second Question

<a href="http://www.codecogs.com/eqnedit.php?latex=\int \frac{x}{2x@plus;1} dx" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\int \frac{x}{2x+1} dx" title="\int \frac{x}{2x+1} dx" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=let u = 2x@plus;1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?let u = 2x+1" title="let u = 2x+1" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=du=2x@plus;1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?du=2x+1" title="du=2x+1" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{du}{dx}=2" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{du}{dx}=2" title="\frac{du}{dx}=2" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{du}{2}=dx" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{du}{2}=dx" title="\frac{du}{2}=dx" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=2x@plus;1=u" target="_blank"><img src="http://latex.codecogs.com/gif.latex?2x+1=u" title="2x+1=u" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=2x=u-1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?2x=u-1" title="2x=u-1" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=x=\frac{u-1}{2}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x=\frac{u-1}{2}" title="x=\frac{u-1}{2}" /></a>

Sub all of this into the original

<a href="http://www.codecogs.com/eqnedit.php?latex=\int \frac{x}{2x@plus;1}\cdot \frac{du}{2}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\int \frac{x}{2x+1}\cdot \frac{du}{2}" title="\int \frac{x}{2x+1}\cdot \frac{du}{2}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{2}\int \frac{\frac{u-1}{2}}{u} du" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{2}\int \frac{\frac{u-1}{2}}{u} du" title="\frac{1}{2}\int \frac{\frac{u-1}{2}}{u} du" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{2}\int\frac{u-1}{2u}du" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{2}\int\frac{u-1}{2u}du" title="\frac{1}{2}\int\frac{u-1}{2u}du" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{4}\int\frac{u-1}{u}du" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{4}\int\frac{u-1}{u}du" title="\frac{1}{4}\int\frac{u-1}{u}du" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{4}[1-\frac{1}{u}]" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{4}[1-\frac{1}{u}]" title="\frac{1}{4}[1-\frac{1}{u}]" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{4}[u-logu]" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{4}[u-logu]" title="\frac{1}{4}[u-logu]" /></a>

sub u back
<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{1}{4}[2x@plus;1-log(2x@plus;1))" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{1}{4}[2x+1-log(2x+1))" title="\frac{1}{4}[2x+1-log(2x+1))+C" /></a>

Which is the same answer as wolframalpha, just rearranged
*And plus, there's +C at the end, but for some reason, I can't change it now*

Fixed before I came back from school.

integrating log (1 Viewer)

jnney

lemon

4025808

Well-Known Member

Shadowdude

Cult of Personality

SpiralFlex

Well-Known Member

such_such

Member

SpiralFlex

Well-Known Member

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