Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 9 / c

The fifth power of the odd function $\tanh x$ is odd and continuous. Its integral over the symmetric interval is therefore $\boxed{{\int }_{-1000}^{1000}{\tanh }^{5}xdx=0}$.

The derivative of $\ln \cosh x$ is $\tanh x$. Since $\cosh (\ln 2)=(2+1/2)/2=5/4$,