2
$\begingroup$

I am supposed to solve a definite integral which involves $\ln$ and $e$. Nowhere in my textbook can I even find examples of how this would be done.

I do know that $\ln(e(x))$ and $e(\ln(x))$ evaluates to $x$, but I don't know how this is supposed to help me with this question.

Determine the possible value of the definite integral $\int_{e}^{e^2} \frac{\ln(\ln(t))}{t\ \log_{2}(w)}dt$

  • 0
    @MichaelFrey By letting $u'(t)=\ln t,\;v(t)=\ln t,\;v'(t)=\dfrac{1}{t}$2012-09-10

1 Answers 1

3

Let $u = ln(t)$. Then $du = \frac{dt}{t}$. Let $c = \frac{1}{log_{2}(w)}$

Then $\int_{e}^{e^2} \frac{\ln(\ln(t))}{t\ \log_{2}(w)}dt = c\int_{1}^{2} \ln(u)du = c[u \ln(u) - u]_{1}^{2} = c(2\ln(2) - 1)$