# Indefinite integral of 1/x (antiderivative of 1/x) (video)

Calculus 1 – Full College Course
Calculus 1 – Full College Course

## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB > Unit 6

Lesson 9: Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals

Indefinite integral of 1/x

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.

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• What is the antiderivative of a constant to the power of negative one? Like 2^-1? Is it ln( l2l ) or something similar?(14 votes)
• A constant to the power of a constant, a constant divided or multiplied by a constant, a constant added to or subtracted from a constant, etc. is still just a constant.
If the value doesn’t change, when you change x, you have a constant.
The graph of 2^-1 (which is 1/2 or 0.5) just like all constants is a straight horizontal line (It doesn’t change with x).
The antiderivative of a straight horizontal line is a line with a slope.
i.e. integral(k dx)= k * x+C
Just to prove it works here:
remember d/dx(0.5x+c) = 0.5 = 2^-1
• What is the difference between lnx and ln|x| ?
• Suppose`x`is a real number. We denote by`|x|`the absolute value of`x`. If`x ≥ 0`, we have`|x| = x`, and if`x < 0`, we have`|x| = -x`.
Denote the natural logarithm of a positive real number`x`by`log(x)`. The function defined by`log(|x|)`is the composition of the logarithm function with the absolute value function. In other words, it is the function defined by`log(|x|) = log(x)`if`x > 0`, and`log(|x|) = log(-x)`if`x < 0`.(32 votes)
• Why is ∫ 1/x dx not equal to, say, ln |2x|, or ln |3x|? For both of these, d/dx = 1/x, right?(10 votes)
• All of that is covered by the`+C`. Remember that`+C`means ANY constant, not just a particular constant.`ln (kx) + C = ln (x) + ln (k) + C`
The`+C`absorbs all of the constants, and`ln (k)`is a constant, thus:`= ln (x) + C`(24 votes)
• At: Why do we search for ln |x|? at that moment? 1:48(7 votes)
• ln|x| includes both positive and negative values as the domain, while lnx does not.(26 votes)
• Is absolute value same as mod function?(3 votes)
• No. They are not the same.
Absolute value means the same thing the distance from 0.
Mod is short for modulo. The modulo operation means the remainder of a division.
Thus:
6 mod 3 = 0
7 mod 3 = 1
8 mod 3 = 2
9 mod 3 = 0
Whereas
| – 9 | = 9
and
| 2 + 3 𝑖 | = √13
NOTE: Your confusion is coming from the fact that the absolute value is also called the modulus. But that is not the same as the modulo (which is what mod stands for). Also note that the term modulus has other uses in mathematics.(13 votes)
• I’ve searched the antiderivative of x^-1, and it says it is log(x)+c not ln(x)+c. Why is this so?(1 vote)
• Professional mathematicians use the natural log, not the common log, as the assumed log. Most professional mathematicians do not use the notation “ln” for the natural log. Thus, at this level of study “log” without a base specified means the natural log.
Therefore,`log (x) + C`and`ln (x) + C`mean the same thing.
Note: it is very unusual to use any other base for a log in calculus than base e. There are a few areas of study where the binary (base 2) log is used, but other than those we nearly always we base e — the math is just much easier with e as the base.(12 votes)
• what about functions like ln(kx) + C, where k is a constant? Wouldn’t that also be a valid antiderivative of 1/x?(3 votes)
• Technically, ln(kx) + C is equivalent to ln(x) + C.
ln(kx) + C
= ln(x) + ln(k) + C
• If my x is negative for 1/x, can’t I just factor out -1 and find the integral of 1/x, which is log(x)(3 votes)
• Of course! And the answer, as you say, would be (-1)ln(x) + C(3 votes)
• ATSal says the derivative of ln|x| is 1/x for all except o but we can clearly see that the derivative are varied, for x> 0 its 1/x and for x<0 its -(1/x). But how does Sal say its the same derivative ? 6:55(2 votes)
• When`x < 0`, we also have`1/x < 0`, so we do indeed have`d/dx log |x| = 1/x`for every real number`x ≠ 0`.(4 votes)
• what does absolute value of x mean?(3 votes)
• In Simple Language-
Its the magnitude of x regardless of its sign
if x= -2 then |x| =2
if x= 1 then |x| = 1
In Mathematical language-
|x| = -x if x<0 and x if x>0
This means that when x<0(i.e. negative) we take its negative and since its already negative it becomes positive.
If x>0 then we take its positive and since its already a positive no. it still remains positive.