Evaluate the following integrals. int_1^{e^2}frac{(ln x)^5}{x}dx

Evaluate the following integrals. int_1^{e^2}frac{(ln x)^5}{x}dx

Question
Integrals
asked 2021-01-02
Evaluate the following integrals.
\(\int_1^{e^2}\frac{(\ln x)^5}{x}dx\)

Answers (1)

2021-01-03
\(\text{Given:}\)
\(\int_1^{e^2}\frac{(\ln x)^5}{x}dx\)
\(\text{Use substitution to solve}\)
\(\text{Let}\)
\(\ln x=t\)
\(\frac{1}{x}dx=dt\)
\(\text{when}\)
\(x=1,\ t=0\)
\(x=e^2,\ t=2\)
\(=\int_0^2 t^5dt\)
\(=[\frac{t^6}{6}]_0^2\)
\(=\frac{1}{6}(2^6-0)\)
\(=\frac{64}{6}\approx10.66\)
0

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