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

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

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

### Relevant Questions

Evaluate each of the following integrals.
$$\int\frac{e^{x}}{1+e^{x}}dx$$
Evaluate the following iterated integrals. $$\int_1^3\int_1^2(y^2+y)dx\ dy$$
Evaluate the following iterated integrals.
$$\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx$$
Evaluate the following integrals.
$$\int_{\ln2}^{\ln3}\frac{e^x+e^{-x}}{e^{2x}-2+e^{-2x}}dx$$
Evaluate the following integrals. Include absolute values only when needed.
$$\int\frac{\ln^2x+2\ln x-1}{x}dx$$
Evaluate the following iterated integrals.
$$\int_1^2\int_0^1(3x^2+4y^3)dydx$$
$$\int_5^{3\sqrt5}\frac{dx}{\sqrt{x^2-9}}$$
$$\int\frac{e^x+e^{-x}}{e^x-e^{-x}}dx$$
$$\int\frac{e^{\sin x}}{\sec x}dx$$
Find the indefinite integral $$\int \ln(\frac{x}{3})dx$$ (a) using a table of integrals and (b) using the Integration by parts method.