1. Evaluate \( \displaystyle{\int \frac{1}{\sqrt{x^2+2}} \, \mathrm{d}x}\).
2. Evaluate \(\displaystyle \int
\frac{16}{x^3-4x} \, \mathrm{d}x\).
3. The region under the graph of \(y
= \sin x\), \(0 \le x \le \pi\), is rotated 360 degrees about
the \(y\)-axis forming a solid of revolution \(S\). Find
the volume of \(S\).
4. Solve the differential equation \(\frac{dy}{dx} = y^2x^2 + y^2\), obtaining an explicit solution.
5. Show that the series \(\displaystyle \sum_{n=2}^\infty \frac{1}{n(\ln n)^2} \) converges.
6. Find the Maclaurin Series for
\(\displaystyle f(x) = \frac{x}{3+x^2} \). What is the
radius of convergence?
7. Determine the
radius and interval of convergence of the power series
\(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n 2^n} (x
- 1)^{n}\).
8. Suppose that \((x,y)\) has distance \(r\ge 1\) from
the origin. Use polar coordinates to show that
9. Find an equation of the line tangent to the curve \(x = 1 + \ln t, y = t^2 + 2\) at the point \((1, 3)\).
10. Find the length of the curve \(x =
e^t -t, y = 4e^{t/2}\), \(0 \le t \le 2\).