Evaluate the integral
_{}(x+sin^{2} x) sin x dx

_{}(x+sin^{2} x) sin x dx
= -x cos x + sin x - cos x +(1/3)cos^{3}x + C

Evaluate the integral
_{}x^{2}/(x^{2}-4) dx

_{}x^{2}/(x^{2}-4) dx
= x + ln|x-2| - ln|x+2| + C

Find the solution of the differential equation y' =
(xy+3x)/(x^{2}+5)
that satisfies the initial condition y(2)=3.

y = 2(x^{2}+5)^{1/2} - 3

1. Consider the sequence {a_{n}} =
{(3n^{2}+7)/(n^{2} + 3n+1)} and the series _{}a_{n} = _{}(3n^{2}+7)/(n^{2} + 3n+1).

(a) Determine if the sequence {a_{n}} converges or diverges. If
the sequence converges, find the limit.

The sequence converges to 3.

(b) Can you use your answer to part (a) to determine if the series
_{}a_{n} converges or
diverges? Explain.

Since the terms of the series do not tend to zero as n approaches
infinity, the series diverges by the n-th term test.

2. Consider the series _{}(-1)^{n-1}
5^{n}/3^{2n}.

(a) Write out the fourth partial sum, s_{4}, of this series.

s_{4} = 5/3^{2} - 5^{2}/3^{4} +
5^{3}/3^{6} - 5^{4}/3^{8}

(b) Determine if this series converges or diverges. If the series
converges, find its sum.

This is a geometric series with first term a = 5/9 and ratio r = -5/9.

The series converges to a/(1-r) = 5/14.

Find the radius of convergence and interval of convergence of the power
series _{}(x-4)^{n} / n
5^{n}

The radius of convergence is R = 5,
and the interval of convergence is [-1,9).

Find the Taylor series of the function f(x) = 1/x^{2} centered at
a = 1.

The Taylor series is _{}(-1)^{n}(n+1)(x-1)^{n} =
1 - 2(x-1) + 3(x-1)^{2} - 4(x-1)^{3} + ...

Sketch the graph of the curve r = 1-sinq
in polar coordinates.

Find the slope of the line tangent to the curve
r = 1-sinq at
q = p.

The slope of the tangent line is equal to 1.

1. Find an equation of the ellipse with foci (±4,0),
and vertices (±5,0).

An equation for this ellipse is x^{2}/25 + y^{2}/9 = 1.

2. Show that x^{2}+y^{2}+z^{2}+2x-6z+5=0 is the
equation of a sphere, and find its center and raduis.

The center is (-1,0,3) and the radius is equal to the square root of 5.