What is the indefinite integral of 6/(x ln(2x)??
6ln(ln(2x))+C
Let u = ln(2x)
du = [2/(2x)] dx = dx/x
6 dx/(x ln(2x)) = 6/u du
= 6 ln u
= 6 ln(ln(2x)) (+ arbitrary constant C)
I did the integration in the next to last line. It does not equal the preceding line.
To find the indefinite integral of 6/(x ln(2x)), we can use the method of integration known as u-substitution.
1. Let's start by identifying the function within the integral that we will substitute. In this case, it is ln(2x).
2. Let u represent ln(2x). We can find du (the differential of u) by differentiating ln(2x) with respect to x:
du/dx = 1/(2x) * 2 = 1/x.
Therefore, du = 1/x dx.
3. Now, let's rewrite the integral substituting u and du:
∫ (6/(x ln(2x))) dx = ∫ (6/u) du.
4. Simplify the expression:
∫ (6/u) du = 6 ∫ (1/u) du.
5. Integrate the simplified expression:
∫ (1/u) du = 6 ln |u| + C,
where C is the constant of integration.
6. Replace u with its original expression:
6 ln |u| + C = 6 ln |ln(2x)| + C.
Therefore, the indefinite integral of 6/(x ln(2x)) is 6 ln |ln(2x)| + C.