\(\text{Given:}\)

\(\int_1^{e^2}\frac{(\ln x)^5}{x}dx\)

\(\text{Use substitution to solve}\)

\(\text{Let}\)

\(\ln x=t\)

\(\frac{1}{x}dx=dt\)

\(\text{when}\)

\(x=1,\ t=0\)

\(x=e^2,\ t=2\)

\(=\int_0^2 t^5dt\)

\(=[\frac{t^6}{6}]_0^2\)

\(=\frac{1}{6}(2^6-0)\)

\(=\frac{64}{6}\approx10.66\)

\(\int_1^{e^2}\frac{(\ln x)^5}{x}dx\)

\(\text{Use substitution to solve}\)

\(\text{Let}\)

\(\ln x=t\)

\(\frac{1}{x}dx=dt\)

\(\text{when}\)

\(x=1,\ t=0\)

\(x=e^2,\ t=2\)

\(=\int_0^2 t^5dt\)

\(=[\frac{t^6}{6}]_0^2\)

\(=\frac{1}{6}(2^6-0)\)

\(=\frac{64}{6}\approx10.66\)