Indefinite Integrals: Solving Homework Problems

In summary, the conversation is about solving an indefinite integral and understanding the difference between log and ln. Specifically, the conversation discusses the integral of 15/(3x+1) dx and determines that its antiderivative is 15ln(3x+1). However, there is a discussion about whether to use log or ln, and it is concluded that they are essentially the same and it is safer to use ln. The final answer is determined to be 5ln|3x+1|.
  • #1
antinerd
41
0

Homework Statement



Hey guys, I'm trying to teach myself how to integrate an indefinite integral.

I just am wondering what you can do with something like this:

Homework Equations



[tex]\int[/tex] 15/(3x+1) dx

The Attempt at a Solution



I'm trying to figure out how to go backwards, but I don't see what terms, when derived, give you 15/3x+1 dx.

Does anyone know a good way to quickly solve these sorts of problems?
 
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  • #2
What about log(3x+1)?
 
  • #3
Dick said:
What about log(3x+1)?

Ah. But wouldn't it have to be:

15 ln(3x+1)

because its derivative would be:

15/(3x+1)

Correct?

So wouldn't the integral of the original problem be just:

[tex]\int[/tex] 15/(3x+1) dx = 15ln(3x+1)

Does it matter whether ln or log is used?
 
  • #4
15 is wrong, do the differentiation again and don't forget the chain rule. it doesn't matter whether it's log or ln because I mean natural logarithm by both. If you want to use logs to a different base then you'll have to adjust the coefficient. log(base a)x=ln(x)/ln(a).
 
  • #5
OK, thanks, let me work this out.
 
  • #6
Ok, what about this thing:

[tex]\int[/tex] 15/(3x+1) dx
and if I factor out the 15:
15[tex]\int[/tex] (3x+1) dx

Now, does the constant just disappear?

and the antiderivative of 1/(3x+1) is just

log (3x+1)
?
 
  • #7
The derivative of log(3x+1) is 3/(3x+1). That's the answer now YOU tell me why.
 
  • #8
Dick said:
The derivative of log(3x+1) is 3/(3x+1). That's the answer now YOU tell me why.

Because if you take the derivative as such:

log(3x+1) dx

You will get

d/dx f(g(x)) = f`(g(x))(g`(x))

Which means that:

d/dx log(3x+1) = (1/(3x+1)) (3)

= 3/(3x+1)

But so now do I have to place a constant to make the derivative 15? I'm wondering if

[tex]\int[/tex]15/(3x+1) = just log(3x+1)

Shouldn't it be 5(log(3x+1)), to give it a 15 on top?
 
  • #9
Exactly, 5*log(3x+1).
 
  • #10
I think it's safer to use ln... some people use log to refer to base-10 logarithm by default...

Also, antiderivative of 1/x = ln|x| (absolute value)

So your answer would be 5*ln|3x+1|
 

FAQ: Indefinite Integrals: Solving Homework Problems

What is an indefinite integral?

An indefinite integral is the reverse process of differentiation. It is a mathematical operation that involves finding a function whose derivative is equal to the given function. It is represented by the symbol ∫ and is commonly used in calculus.

How do I solve indefinite integrals?

To solve an indefinite integral, you can use various techniques such as substitution, integration by parts, and partial fractions. It is important to identify the type of integral and choose the appropriate method to solve it. Practice and understanding of basic rules and formulas are also essential in solving indefinite integrals.

What are the common mistakes to avoid when solving indefinite integrals?

Some common mistakes to avoid when solving indefinite integrals include forgetting to add the constant of integration, making algebraic errors, and incorrect application of integration rules. It is crucial to double-check your work and be mindful of the rules and steps involved in solving indefinite integrals.

How can I check if my solution to an indefinite integral is correct?

You can check your solution to an indefinite integral by taking the derivative of your answer and verifying if it is equal to the original function. If they are equal, then your solution is correct. You can also use online integration calculators or ask a classmate or teacher to check your work.

What are the applications of indefinite integrals?

Indefinite integrals have various applications in mathematics, physics, and engineering. They can be used to find the area under a curve, calculate displacement, and solve problems involving rates of change. They are also used in the development of mathematical models and in areas such as optimization and data analysis.

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