I would really appreciate your help with 3 integration problems that I am completely unsure about.

The first is (lnx)^2 dx

The Second is e^x sin(3x) dx

The third is a definitive integral between pi and zero, sin^3(x) dx

Thank you!

\(\displaystyle \int{(\ln{x})^2\,dx} = \int{\ln{x}\cdot\ln{x}\,dx}\).

Using integration by parts with

\(\displaystyle u = \ln{x}\) so that \(\displaystyle du = \frac{1}{x}\)

\(\displaystyle dv = \ln{x}\) so that \(\displaystyle v = x\ln{x} - x\)

we find

\(\displaystyle \int{\ln{x}\cdot\ln{x}\,dx} = \ln{x}(x\ln{x} - x) - \int{\frac{x\ln{x} - x}{x}\,dx}\)

\(\displaystyle =x(\ln{x})^2 - x\ln{x} - \int{\ln{x} - 1\,dx}\)

\(\displaystyle =x(\ln{x})^2 - x\ln{x} - (x\ln{x} - x - x) + C\)

\(\displaystyle = x(\ln{x})^2 - x\ln{x} - x\ln{x} + 2x + C\)

\(\displaystyle = x(\ln{x})^2 - 2x\ln{x} + 2x + C\).