Indefinite Integrals?

Question

S(3dx/(xln5x)) (where S is the integration sign)

What to do? I know integration is "backwards differentiation" ... but how do you go about it with a division.

Answer 1

Make the sub
u = ln(5x)
du = dx/x

So integral transforms to
3*du / u
which integrates to
3*ln(Cu)
= 3*ln[C*ln(5x)]

where C = constant

Answer 2

I prefer to write the question with the dx at the end; i.e. like this:

∫ (3/ [xln(5x)] dx )

We have to solve this using substitution. But first, I'm going to rewrite this such that we have (1/x) dx at the end.

∫ ( 3/ln(5x) (1/x) dx )

Now, we use substitution.

Let u = ln(5x). Then
du = [ 1/(5x) ](5) dx, or
du = (1/x) dx.
Notice that (1/x)dx is the tail end of our integral; this is going to be replaed by du. Our integral then becomes

∫ (3/u) du

Factoring out the 3 from the integral,

3∫ (1/u) du

And this is now a simple integral.

3 ln|u| + C

But u = ln(5x), so our final answer is

3 ln| ln(5x) | + C

Answer 3

try using substitution with u = ln x

therefore du = dx /x and it becomes the integral of du/u

Answer 4

We'll do this by substitution:
Let u = ln 5x. Then du = 5dx/5x = dx/x
So you have
3∫ du/u = 3 ln |u| + C = 3 ln | ln |5x| | + C