my professor said to use trigonometric substitution for int (ln^3 w)/(w sqrt(ln^2 w)) dw

[Jurt]

I've been taking calculus, and I'm having a hard time with this question. My professor told us to use Trigonometric Substitution and Im having trouble doing it.

 

[stapel]

I've been taking calculus, and I'm having a hard time with this question. My professor told us to use Trigonometric Substitution and Im having trouble doing it.View attachment 37172

What are your thouights? What have you tried? How far have you gotten? ("Read Before Posting")

When you reply, please state which of the following is the argument of the square root:

[imath]\qquad \textrm{(a) } \ln^2(x) - 4[/imath]

[imath]\qquad \textrm{(b) } \ln^2(x - 4)[/imath]

Thank you!

 

[Dr.Peterson]

I've been taking calculus, and I'm having a hard time with this question. My professor told us to use Trigonometric Substitution and Im having trouble doing it.

View attachment 37172

Presumably [imath]\ln^3w[/imath] is intended to mean [imath]\left(\ln(w)\right)^3[/imath], not [imath]\ln(\ln(\ln(w)))[/imath], which is a more standard meaning for powers of functions in general. (We usually, in my experience, put exponents only on trig functions with the former meaning.)

I would take this in two steps, first doing a "u-substitution" with [imath]\ln(w)[/imath].

 

[BigBeachBanana]

I've been taking calculus, and I'm having a hard time with this question. My professor told us to use Trigonometric Substitution and Im having trouble doing it.View attachment 37172

Assuming the denominator is the case (a) in post#2, let [imath]u = \ln^2(w)-4[/imath].