Indefinite Integrals: Solving Homework Problems

[antinerd]

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

\int 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?

 

[Dick]

What about log(3x+1)?

 

[antinerd]

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:

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

Does it matter whether ln or log is used?

 

[Dick]

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).

 

[antinerd]

OK, thanks, let me work this out.

 

[antinerd]

Ok, what about this thing:

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

Now, does the constant just disappear?

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

log (3x+1)
?

 

[Dick]

The derivative of log(3x+1) is 3/(3x+1). That's the answer now YOU tell me why.

 

[antinerd]

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

\int15/(3x+1) = just log(3x+1)

Shouldn't it be 5(log(3x+1)), to give it a 15 on top?

 

[Dick]

Exactly, 5*log(3x+1).

 

[learningphysics]

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|