integrate (3dx/xln3x)
Try the substitution
u=ln(3x)
du= 3dx/(3x)=dx/x
...
integrate 3dx
To integrate the given function ∫(3dx / x ln(3x)), we can use a combination of substitution and integration by parts.
Let's start by making a substitution. Let u = ln(3x). This implies du/dx = 1/x. Solving for dx, we get dx = du / (1/x) = x du.
Substituting the new variables, the integral becomes:
∫(3dx / x ln(3x)) = ∫(3x du / x ln(3x))
= 3 ∫(du / ln(3x))
Next, we use another substitution. Let v = 3x. Therefore, x = v/3 and dx = (1/3) dv.
Applying the new variables, the integral becomes:
3 ∫(du / ln(3x)) = 3 ∫(du / ln(v))
To simplify further, we can pull out the constant factor of 3 out of the integral:
3 ∫(du / ln(v)) = 3 * (1/ln(v)) ∫du
= 3u / ln(v) + C
Now, we substitute back the original variables:
= 3ln(3x) / ln(3x) + C
= 3 + C
Therefore, the integral of (3dx / x ln(3x)) is 3 + C, where C is the constant of integration.