Show that \(\displaystyle \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\left((\sin\theta+\cos\theta)^2+(\sin\theta-\cos\theta)^2\right)\,\mathrm{d}\theta=k\pi\), where \(k\) is an integer to be found.

Find \(\int_{0}^{2}\left(1+\mathrm{e}^{2 x}\right)^{2} \mathrm{~d} x\), giving your answer in exact form.

Find the exact value of

\(\displaystyle \int_3^5 \frac{(x-1)^2}{x^3}\,\mathrm dx.\)

Find the exact value of

\(\displaystyle \int_2^3\frac{(x+2)^2}{x}\,\mathrm dx.\)

Find

\(\int_3^5\left(\frac1{x-1}-\frac1{(x-1)^2}\right)\,dx,\)

giving your answer in the form \(a+\ln b\), where \(a\) and \(b\) are rational numbers.