Evaluate the integral of the natural logarithms.

Question

1. ∫(e to 6) of dx/(xlnx)

I started off by separating dx so that it was (1/xlnx)dx and I used the ∫1/x(dx)=ln|x|+C identity so that the equation was ln|xlnx|+C but I'm not sure if that was correct or how to proceed.

1. ∫(1 to 2) of dt/(8-3t)

Answer 1

Actually if you use u = ln x, then du = ¹/_x dx and u(e) = ln e = 1, u(6) = ln 6

1. so ∫e⁶ of dx/(x ln x) = ∫₁^{ln6 1}/_u du = ln u|₁^ln6 = ln(ln6)-ln1 =ln(ln 6)
2. the second integral involves using u = 8-3t so be careful with the negative signs