DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1063

FINAL EXAMINATION
APRIL 1999
TIME: 3 HOURS

ANSWER ALL QUESTIONS WITHOUT THE USE OF A CALCULATOR. CREDIT WILL BE GIVEN FOR PRESENTATION AND METHODS OF SOLUTION.

1. Evaluate the following integrals:
(a) ${\displaystyle \int\;\frac{\cos x}{1 + sin^{\textstyle 2} x}\;dx\;;}$ (b) ${\displaystyle \int\;(x^{\textstyle 2} + x + 2)\ln x\;dx\;;}$
(c) ${\displaystyle \int\;\cos^{\textstyle 5}x\;dx\;;}$ (d) ${\displaystyle \int\;\frac{2x+1}{x^{\textstyle 2} + 4x + 8}\;dx\;;}$
(e) ${\displaystyle \int\;\frac{1}{x + \sqrt{x}}\;dx\;;}$ (f) ${\displaystyle \int\;\tan^{\textstyle 4}x\;dx\;;}$
(g) ${\displaystyle \int\;x^{\textstyle 3}(2-x^{\textstyle
2})^{\textstyle 1/5}\;dx\;;}$ (h) ${\displaystyle \int\;e^{\textstyle x} \sec(e^{\textstyle x}+3)\;dx\;.}$
2. Evaluate:
(i) ${\displaystyle \int\;\frac{x^{\textstyle 2} - x + 6}
{(2x+1)(x-1)^{\textstyle 2}}\;dx\;;}$ (ii) ${\displaystyle \int\;\frac{1}{x^{\textstyle 2}
\sqrt{x^{\textstyle 2} + 9}}\;dx\;;}$
(iii) ${\displaystyle \int_{\textstyle 0}^{\textstyle 2} \vert x^{\textstyle
2}-1\vert\;dx\;.}$
3. Evaluate the following limits and improper integrals, or show that they do not exist:
(a) ${\displaystyle \lim_{x \rightarrow 0}\;\frac{3e^{\textstyle 2x}
- 2e^{\textstyle 3x} - 1}{x \sin x}\;;}$ (b) ${\displaystyle \lim_{x \rightarrow - \infty} xe^{\textstyle x}\;;}$
(c) ${\displaystyle \int_{\textstyle 3}^{\textstyle
\infty}\;\frac{1}{x^{\textstyle 2} - 4}\;dx\;;}$ (d) ${\displaystyle \int_{\textstyle 0}^{\textstyle
\pi/2}\;\frac{\sin x}{\cos x \sqrt{\cos x}}\;dx\;.}$
4. (i) Use the definition of the Riemann integral, with a regular partition of the interval [-1,2] and the choice ci = xi in each subinterval [xi-1,xi], to evaluate
\begin{displaymath}\int_{\textstyle -1}^{\textstyle 2} (2x^{\textstyle 2} - 2x -
1)\;dx. \end{displaymath}
(ii) Compute the derivative: ${\displaystyle
\frac{d}{dx} \int_{\textstyle 3x}^{\textstyle
2}\;\frac{t}{1+t^{\textstyle 3}}\;dt}$.
5. Sketch the finite region R in the x-y plane that is bounded by the two curves, y = x3 and x = y2 Hence express the following measures as integrals in the stated variables, but do not evaluate the integrals:
(a) the area of R as an integral in x;
(b) the volume of the solid, generated on rotation of R about the y-axis, as an integral, (i) in x; and (ii) in y;
(c) the volume of the solid, generated on rotation of R about the line x = 1, as an integral in y.
6. Solve:
(i) ${\displaystyle \frac{dy}{dx} = (x + 1)e^{\textstyle x+y}\;;}$
(ii) ${\displaystyle \frac{dy}{dx} + (\cot x)y = \sin x,\;\;y \left(
\frac{\pi}{2} \right) = \pi\;;}$
(iii) ${\displaystyle \frac{dy}{dx} + \frac{y}{x} = \frac{x}{y}\;.}$
7. (a) Express the following complex numbers in the form a + ib, where a and b denote real numbers:
${\displaystyle z_{\textstyle 1} = \frac{3-i}{2 + 3i}}$ ; ${\displaystyle z_{\textstyle 2} = (1 + i)^{\textstyle
6}\;;}$
${\displaystyle z_{\textstyle 3} = e^{\textstyle 2-i \pi/6}}$ ; ${\displaystyle z_{\textstyle 4} = \sin(2-5i)\;.}$
(b) Find all solutions of the equation, $z^{\textstyle 4} =
-\sqrt{3} + i$.
(c) Use DeMoivre's theorem to determine the formula for $\sin 3
\theta$ as a sum of powers of $\sin \theta$.