Week 7 Exercises

7.5 Sheet 15 - Integration using Partial Fractions

  1. \(\displaystyle\int \frac{2x+3}{x^2+x-30}\ dx\)
  2. \(\displaystyle\int \frac{17-7x}{(3-x)(2-x)}\ dx\)
  3. \(\displaystyle\int \frac{5}{x^2-1}\ dx\)
  4. \(\displaystyle\int \frac{x^2-3x+3}{(x-1)(x-2)(x-3)}\ dx\)
  5. \(\displaystyle\int \frac{2x^2-2x-2}{x^3-2x^2-x+2}\ dx\)
  6. \(\displaystyle\int \frac{7}{x^2-4}\ dx\)
  7. \(\displaystyle\int \frac{5x+2}{x^2-4x+4}\ dx\)
  8. \(\displaystyle\int \frac{1}{4x^2-9}\ dx\)
  9. \(\displaystyle\int \frac{1}{x^2-4x-8}\ dx\)
  10. \(\displaystyle\int \frac{1}{x^2-4x-12}\ dx\)
  11. \(\displaystyle\int \frac{\cos(x)}{\sin(x)(1+\sin(x))}\ dx\)
  12. \(\displaystyle\int \frac{\sin(x)}{1-4\cos^2(x)}\ dx\)
  13. \(\displaystyle\int \frac{e^x}{1-e^{2x}}\ dx\)
  14. \(\displaystyle\int \frac{1}{x^2-2x-1}\ dx\)

{Solutions: 1. \(\frac{9}{11}\ln|x+6| + \frac{13}{11}\ln|x-5|+C\); 2. \(-3\ln|x-2|-4\ln|x-3| + C\); 3. \(\frac{5}{2}(\ln|x-1|-\ln|x+1|)+C\); 4. \(\frac{7}{2}\ln|x-1| -13\ln|x-2|+\frac{21}{2}\ln|x-3|+C\); 5. \(\frac{1}{3}\ln|x+1|+\ln|x-1|+\frac{2}{3}\ln|x-2|+C\); 6. \(\frac{7}{4}(\ln|x-2|-\ln|x+2|) +C\); 7. \(5\ln|x-2| - 12/(x-2) +C\); 8. \(\frac{1}{12}(\ln|2x-3|-\ln|2x+3|)+C\); 9. \(\frac{1}{4\sqrt{3}}\left( \ln|x-2\sqrt{3}-2| - \ln|x+2\sqrt{3}-2| \right)+C\); 10. \(\frac{1}{8}(\ln|x-6|-\ln|x+2|) +C\); 11. \(\ln|\sin(x)| - \ln(\sin(x)+1) + C\); 12. \(\frac{1}{4}\left( \ln|2\cos(x)-1| -\ln|2\cos(x)+1| \right) +C\); 13. \(\frac{1}{2}\left( \ln(e^x+1) - \ln|e^x-1| \right) + C\); 14. \(\frac{1}{2\sqrt{2}}\ln\left|\frac{x-1-\sqrt{2}}{x-1+\sqrt{2}}\right| +C\); }

7.6 Sheet 16 - Definite Integration

Exercise 1

Determine the values of the following definite integrals

  1. \(\int_0^2 x^2\ dx\)
  2. \(\int_0^3 (4-x)^2\ dx\)
  3. \(\int_0^4 3\sqrt{x}\ dx\)
  4. \(\int_1^2 t^2+3t\ dt\)
  5. \(\int_0^{\pi/2}4\cos(\theta) \ d\theta\)
  6. \(\int_1^2 \cos(\theta)-\sin(\theta)\ d\theta\)
  7. \(\int_1^9 1/\sqrt{p}\ dp\)
  8. \(\int_1^2 r^3-1/r\ dr\)
  9. \(\int_2^3 1/x\ dx\)
  10. \(\int_1^4 (x+1)/(2\sqrt{x})\ dx\)
  11. \(\int_2^83/x \ dx\)
  12. \(\int_{0.5}^1(e^t+e^{-t})/2 \ dt\)
  13. \(\int_1^4 \sqrt{t}(1+t)^2 \ dt\)
  14. \(\int_{0.2}^{0.5} \cos(3x)\ dx\)
  15. \(\int_3^4 (x+1)(2-x)\ dx\)
  16. \(\int_{\pi/6}^{\pi/4} \sec^2(\theta)\ d\theta\)

{Solutions: 1. \(8/3\); 2. \(21\); 3. \(16\); 4. \(41/6\); 5. \(4\); 6. \(\sin(2)+\cos(2)-\sin(1)-\cos(1) \approx -0.889\) (\(3\) d.p.); 7. \(4\); 8. \(15/4 - \ln(2) \approx 3.057\) (\(3\) d.p.); 9. \(\ln(3/2) \approx 0.405\) (\(3\) d.p.); 10. \(10/3\); 11. \(3\ln4 \approx 4.159\) (\(3\) d.p.); 12. \(\approx 0.654\) (\(3\) d.p.); 13. \(6904/105 \approx 65.75\) (\(2\) d.p.); 14. \(\frac{1}{3}(\sin(3/2)-\sin(3/5)) \approx 0.1443\) (\(4\) d.p.); 15. \(-41/6 \approx -6.833\) (\(3\) d.p.); 16. \(1-1/\sqrt{3} \approx 0.423\) (\(3\) d.p.); }

Exercise 2

Evaluate the area bounded by the curve, the x axis and the given ordinates for the following:

  1. \(y=x^2-2x\), \(x=0\) and \(x=2\)
  2. \(y=3x+x^2\), \(x=1\) and \(x=3\)
  3. \(y=\sin(x)\), \(x=0\) and \(x=\pi/3\)
  4. \(y=3\cos(3x)\), \(x=\pi/6\) and \(x=\pi/3\)
  5. \(xy=4\), \(x=3\) and \(x=6\)
  6. \(y=\sin(x)+\cos(x)\), \(x=\pi/6\) and \(x=1\)
  7. \(y=1/e^{2x}\), \(x=0\) and \(x=2\)
  8. \(y=(e^x-e^{-x})/2\), \(x=0\) and \(x=2\)
  9. \(y=4at\), \(t=0\) and \(t=1\) where \(a\) is a constant

{Solutions: 1. \(\approx 1.33\) (\(2\) d.p.); 2. \(\approx 20.67\) (\(2\) d.p.); 3. \(0.5\); 4. \(1\); 5. \(\approx 2.77\) (\(2\) d.p.); 6. \(\approx 0.667\) (\(3\) d.p.); 7. \(\approx 0.491\) (\(3\) d.p.); 8. \(\approx 2.76\) (\(2\) d.p.); 9. \(8a^2\); }