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# 𝘶-substitution: logarithmic function

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.D (LO)
,
FUN‑6.D.1 (EK)
Doing u-substitution with ln(x). Created by Sal Khan.

## Want to join the conversation?

• How would you solve the integral of ln(x)
• From the author:
• whats the derivative of pi? or does it not have one?
• The derivative of any constant is zero. Therefore, the derivative of pi is 0.
• My calculator can't define ln|ln(x)|, is that normal?
• If it can't take it in one step, then simplify the process. First get ln(x), then get it's absolute value, and finally calculate ln of final value. You would need piece of paper to keep track of values.
• I don't understand why you can take out the pi outside of the integral, is it b/c it's a constant? What are other examples of things you can take out in fractions?
• Yes, it is because π is a constant.
Remember that an integral can be interpreted as a limit problem and the ability to factor out constants is a basic property of limits.

However, factoring out the constants is just a matter of convenience. It is not actually required so long as you know what you're doing.
• I don't get why we have to make it an absolute value towards the end around ?
• The antiderivative of 1/x is often given as ln(x) + C but it's more complete to give it as ln(|x|) + C. If you think of the graph of 1/x, you'll realize that it has two segments that are mirror images of each other, one in the first quadrant and the other in the third quadrant, and when we give the antiderivative without the absolute value we're really giving a partial solution that covers only the positive values of x, because ln(x) is undefined for negative x.

In this example, the original problem involves a logarithm, which can never apply to a negative number, but the integral is a log of a log, and the inner log could be negative, which means the more complete solution is one that includes the absolute value.
• Sal will you answer next questions, I hope.
What do dy and dx mean idividually?
What is definition of "dx"?
Is dy/dx a fraction or not fraction?
• nice question
dy/dx is not a fraction but in the field of calculus it is the rate of change of y with respect to x
dx is the slight change in the quantity of x
same is the case of dy
• The Integral of 2/(e^(2x)+4). I have used the substitution u=e^(2x)+4 and get as far as 1/u(u-4)du but I cannot get any further. The answer gives a log function. I hope this does not qualify as a homework question but I have tried everything can think of
• Why does the 'du' go away after doing the integral?
• The common nomenclature of the integral includes both the integral sign at the left and the differential `du` at the right. When you solve the integral both the integral sign at the left and the differential at the right go away. You must consider them to alway go together, so when one goes the other goes as well.
``Integral sqrt(u) du= Integral u^(1/2) du= 2/3*u^(3/2) + C``