DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1053

FINAL EXAMINATION
April 1998
TIME: 3 HOURS

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

1. Evaluate the following integrals:
(a) ${\displaystyle \int \frac{(\tan^{-1}x)^2}{1+x^2}\;dx\;;}$ (b) ${\displaystyle \int (3x^2-4x+1)e^{-x}\;dx\;;}$
(c) ${\displaystyle \int \frac{1}{1+\sqrt{2x+3}}\;dx\;;}$ (d) ${\displaystyle \int \sec^4x\;dx\;;}$
(e) ${\displaystyle \int \sin^5x\;dx\;;}$ (f) ${\displaystyle \int \frac{3}{(2x-1)(5x+4)}\;dx\;;}$
(g) ${\displaystyle \int \cos x \cos 3x\;dx\;;}$ (h) ${\displaystyle \int x^2 \ln(x^2+1)\;dx\;.}$
2. Evaluate:
(a) ${\displaystyle \int \frac{x+1}{\sqrt{4x-x^2}}\;dx\;;}$
(b) ${\displaystyle \int \frac{x^2-x-3}{(2x+1)(x^2+2x+3)}\;dx\;;}$
(c) ${\displaystyle \int \frac{1}{x(x^2-9)^{3/2}}\;dx\;.}$
3. Compute the values of the following expressions:
(a) ${\displaystyle \int_{-1}^3 \vert x^2-x\vert\;dx\;;}$ (b) ${\displaystyle \frac{d}{dx} \int_{2x}^{3x} \sqrt{t^3+1}
\;dt\;;}$
(c) ${\displaystyle \lim_{x \rightarrow 0}\;
\frac{x^2-\cos x + 1}{x \tan x}\;;}$ (d) ${\displaystyle \lim_{x \rightarrow \infty} (x+1)e^{-x}\;.}$
4.(a) State, clearly, the Riemann definition of the integral of a function, f (x), defined over a finite interval $a \leq x \leq b$. Use this definition, with a regular partition of [-1,2] and the choice ci = xi in each sub-interval [xi-1,xi], to evaluate ${\displaystyle \int_{-1}^{2} x^3\;dx}$.
(b) Evaluate each of the following improper integrals, or else show that no value exists:
(i) ${\displaystyle \int_1^e\;\frac{1}{x \sqrt{\ln x}}\;dx\;\;;}$ (ii) ${\displaystyle \int_1^{\infty} \tan^{-1}x\;dx\;.}$
5. Sketch the finite region R in the x-y plane that is bounded by the two curves $y = x^2,\;\;y^2 = x$. 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 y;
(b) the volume of the solid generated on the rotation of R about the x-axis as an integral,
(i) in x; (ii) in y;
(c) the volume of the solid generated on rotating R about the line y = 1 as an integral in x.
6. Solve:
(a) $2(y-4x^2)dx + xdy = 0;\;\;\;y(1) = -1$;
(b) $xdy = (y + \sqrt{x^2-y^2}\;)\;dx$;
(c) x2y' + 2xy = y3.
7.(a) Express each of the following numbers in the form a + ib:
${\displaystyle z_1 = \frac{3+i}{1-2i}}$; $z_2 = e^{1-i\pi/3}\;\;;$
$z_3 = \cos(2-i)$; $z_4 = (\sqrt{3} + i)^6\;\;;$
$z_5 = \log(-3 - 3i)$; $z_6 = \sinh(1-i)\;$.
(b) Prove: $\vert z_1 + z_2\vert \leq \vert z_1\vert + \vert z_2\vert$.
(c) Find all solutions to the equation: $z^4 = 1 - i \sqrt{3}$.