# Logs and antiderivatives

• vipertongn
In summary, the conversation discusses the use of substitution and properties of natural logs to solve the equation (x+2)/(x^2+4x) dx. The correct substitution is u = x^2 + 4x, and the antiderivative is ln |u| + C. This is because the derivative of ln |u| is 1/u, similar to the relationship between y = ln(x) and x = e^y.

#### vipertongn

S (x+2)/(x^2+4x) dx

I've been learning about natural logs and their properties but at this answer I get befuddled at how to work it out. I think perhaps substitution, and some properties with logs but I am very weak in logs...Didn't do well in it when I was in algebra.
I want to know how to do this equation, although I know it has this as an answer:

log(abs(x^2+4x))/2

Thanks for any help.

vipertongn said:
I think perhaps substitution

You think rightly. Now take a stab at choosing the right substitution. There aren't that many choices available, so this shouldn't be too hard.

hmmm ok let's see u = x^2+4x --> du=2x+4 --> du/2=x+2

makes the equation

1/2 S du/U but 1/u means ln|u| though right? why is it a log?

vipertongn said:
hmmm ok let's see u = x^2+4x --> du=2x+4 --> du/2=x+2

makes the equation

1/2 S du/U but 1/u means ln|u| though right? why is it a log?

That's the right substitution, but you're missing something in du.
u = x^2 + 4x ==> du = (2x + 4)dx

$$\int du/u = ln |u| + C$$
An antiderivative of 1/u is ln |u | because the derivative of ln |u| is 1/u. It's a sort of inverse relationship, similar to the relationship between the equations y = ln(x) and x = e^y.

## 1. What is the difference between a log and an antiderivative?

A log, or logarithm, is a mathematical function that is the inverse of an exponent. It tells you what power you need to raise a base number to in order to get a given number. An antiderivative, on the other hand, is the opposite of a derivative and is used to find the original function when given the derivative of that function.

## 2. How are logs and antiderivatives used in real-world applications?

Logs and antiderivatives are used in fields such as physics, finance, and engineering to model and analyze various phenomena. They can be used to calculate growth rates, measure radioactive decay, and predict stock market trends, among other things.

## 3. What is the process for finding the antiderivative of a function?

The process for finding the antiderivative, or integral, of a function involves reversing the steps of differentiation. This can be done using a variety of techniques such as substitution, integration by parts, and partial fractions. It is important to note that the antiderivative is not always a unique function, as it can have an arbitrary constant added to it.

## 4. What are some common properties of logs and antiderivatives?

Some common properties of logs and antiderivatives include the product rule, quotient rule, and chain rule. These rules help simplify the process of finding the derivative or antiderivative of more complex functions. Additionally, both logs and antiderivatives are used to solve exponential equations and can be represented visually on a graph.

## 5. Can logs and antiderivatives be used interchangeably?

No, logs and antiderivatives cannot be used interchangeably. While they may seem similar in some ways, they serve different purposes and have different mathematical properties. Logs are used to find the exponent of a given number, while antiderivatives are used to find the original function when given its derivative.