Silly question here, but I am not getting a certain part in the solution in a problem.
Evaluate the integral of [(sin(ln(39x)))/x]dx. We are supposed to take u=ln(39x), and get du=(1/x)dx, and the then whole thing is simply solved using integration formulas. But my confusion is in the du=(1/x)dx. If u=ln(39x)m shouldnt du=39(1/39x)dx ?? I did this and got an extra -1/39 in my answer, but the solution doesnt have that as they just took (1/x)dx. I wonder why? Am I missing something here?
yes, you are right...almost...
u= ln (39x)
but ln (ab)=lna + lnb
u= ln39 + lnx
du= 0+1/x dx
Ohhh yeah! Got it! Thanks!
In the given problem, you are asked to evaluate the integral of [(sin(ln(39x)))/x]dx by using the substitution method. Let's go through the steps and understand the reason behind the discrepancy you encountered.
1. We start by making the substitution u = ln(39x).
2. To find du, we differentiate both sides of the equation u = ln(39x) with respect to x:
du/dx = (1/39x) * d(39x)/dx.
Simplifying, we get du/dx = (1/x).
Please note that since we took the derivative with respect to x, the derivative of ln(39x) with respect to x is (1/x), not 39(1/39x).
Now, let's integrate the given expression using the substitution method. We have:
∫[(sin(ln(39x)))/x]dx
Using the substitution u = ln(39x), we can rewrite the integral as:
∫sin(u)du
This is now a standard integral. The antiderivative of sin(u) is -cos(u), so we can integrate to get:
-∫cos(u) du
Replacing u with ln(39x):
-∫cos(ln(39x)) du
Now, we can evaluate the integral:
-∫cos(ln(39x)) du = -sin(ln(39x)) + C
This is the final answer, where C is the constant of integration.
To summarize, the substitution was made correctly, and the differential du was correctly obtained as (1/x)dx. The discrepancy might have occurred due to a simple arithmetic mistake where the factor of 39 in the denominator was incorrectly incorporated into the derivative, thus resulting in an extra term in the final answer.
It's always a good idea to check your calculations for possible errors and carefully track the algebraic manipulations during the integration process to avoid such discrepancies.