91Ó°ÊÓ

Find or evaluate the integral. $$ \int \tan ^{5} \frac{x}{2} d x $$

Find the surface area of the ellipsoid formed by revolving the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b\), about the \(x\) -axis.

Find the length of the astroid \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\), where \(a>0\).

Find the values of \(p\) for which the integral \(\int_{0}^{1} 1 / x^{p} d x\) converges and the values of \(p\) for which it diverges.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(f(x) \leq g(x)\) for all \(x\) in \([a, \infty)\) and \(\int_{a}^{\infty} f(x) d x\) converges, then \(\int_{a}^{\infty} g(x) d x\) also converges.