Indefinite integral of 1/x | AP Calculus AB | Khan Academy

In differential _calculus we learned that the derivative of ln(x) is 1/x. Integration goes the other way: the integral (or antiderivative) of 1/x should be a function whose derivative is 1/x. As we just saw, this is ln(x). However, if x is negative then ln(x) is undefined! The solution is quite simple: the antiderivative of 1/x is ln(|x|). Created by Sal Khan.

AP Calculus AB on Khan Academy: Bill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and heÕs part of the teaching team that helped develop Khan AcademyÕs AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.

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[johnholme]

As a physicist studying mathematics, all I’m after is a good intuitive grasp and this video is perfect for my needs!

[AFirongeneral]

I LOVE KHAN ACADEMY!! I WISH I CAN SIT ON MY COMPUTER AND WATCH ALL THE VIDEOS ALL DAY!!!

[someone]

Ooooh now I understand why we take the absolute value of the natural logarithm when doing integrals of (x^-1) or (f'(x)/f(x)) not only because the number inside the natural logarithm has to be positive, but also to make it's domain the same as the domain of (x^-1)

[MohaMMaDiN]

what if we have the function:

1/[2*sqrt(x)] for x>0

the integral of the function is then:

sqrt(x) + C for x>=0

here as you can notice, the domain of both functions has changed !

[Omar-sqzz]

Save yourself time the integral of 1/x is ln|x| don't forget the absolute vale

[SpinnTV]

Thanks a lot . I really love your channel

[sambowwow]

In mathematical logic, integrals estimate the area under the curve of a function. the function 1/x is infinite. why doesn't the answer sums up to infinite?

[rhoadess]

What also blows my mind is that D(e^x) = e^x, a function which is its own slope/derivative! From there we can prove D[ln(x)] = 1/x and from there we can show ∫(1/x)dx = ln(x) + c, but ... wait, something is wrong here, and Sal shows it here. So then shouldn't we say D[ln|x|] = 1/x, rather than D[ln(x)] = 1/x?

[eminemworthy]

nice video khan like always keep it up!

[Tunpredictable]

Please do not pollute the comment section.

[Thegamemakur]

He uses a tablet and learn calculus, it's awesome

[nurlatifahmohdnor]

Page 26 and 27




Sa-li-ngan = 1/x or x to the power of minus 1

[chrisjohnston]

God the ln is so weird. If you had a gravitational force equal to some GMass of body * mass of orbiting object (GMm, where M = mass of body, m = mass of orbiting object), divided by r^2(which would be the radius or distance from center of mass of the body in question, squared) or f = 1/r^2, recalling the definition of work (work = Force * Displacement, and dW = Fdr, you can substitute GMm/r^2 into the equation to get dW = GMm/r^2 * dr, and take the definite integral of the right side from the initial radius (R0) to the radius you're at, (R), you get -GMm/r + GMm/r0, or GMm*(1/r0 - 1/R). You can then evaluate R at infinity to get the total work done moving an object from point r0 in the gravitational field to infinity, and it's a finite value equal to GMm/r0! If you say that the gravitational field is equal to GMm/r (instead of r^2), substitute it into the above (simple) work differential equation you get dW = GMm/r dr, take the definite integral from r0 to r to get the integral of GMm * 1/r dr = GMm * ln|R-R0|. Evalute R at infinity and you get GMm*ln|infinity|. which equals infinity, which means that despite the fact that the gravitational force is tends to 0 as distance approaches infinity, the work needed to get there is STILL INFINITY? What gives?

[elidrissii]

Haha. We wrote our comments at the same time and we both described his comment as "essay".

[YoAirFresh]

The problems on Khan Academy that are associated with this video supply incorrect answers. The answers required are inconsistent with what is taught in this video. I generally report through the site, but this is a widespread bug. Khan Academy does not accept answers containing the absolute value notation such as -4ln|x|. It only accepts -4lnx, which is not entirely correct. Several problems like this need to be fixed.

[HideMyselfFrom]

Amin, If my memory serves me correctly, "natural log" is essentially the same thing as "log base e." So if we're just taking the indefinite integral, then I don't believe it makes a difference.

[johnstamoser]

ln|x| is NOT the MOST general antiderivative of 1/x! ln|x|, where x is any number in the real numbers, can be defined by the piecewise-defined function:

f(x)={ ln(x), x greater than 0
{(ln (-x)), x less than 0

Now, suppose you're given a function F'(x)=1/x, whose antiderivative is f(x). Then the MOST GENERAL antiderivative is:
f(x)={ ln(x)+C, x is greater than 0
{(ln (-x))+D, x less than 0
where C and D are some constants.

I can't believe how common that mistake is made.

[spaghetti]

I'm suprised you didn't discuss all of the technicalities in a 7 minute video. To be precise the integral of 1/x is ln|Cx|+c. There also has to be constant inside of the absolute value because when taking the derivative of my answer, the C's cancel via chain rule. For example, the derivative of ln(2x) is (1/2x)•2 which is also equal to 1/x.

[divermike]

So why is d/dx lnx =1/x? How was that first proven?

[Mean_men]

Can we write 1/x =x^(-1+1)/(-1+1) in intergration?or should we write result as ln|x| .but y not the other way