MATH3503 Online Assignment 8

• Make sure to use the link on the Learning Hub, which contains a user name and a user ID unique to you, otherwise your mark would not be recorded. You may use the URL without '?uname=...&uid=...' if you just want to practice. You will get different numbers every time you open the page.
• Numerical answers are evaluated as correct if the difference from the correct value is less than `0.001 times` (correct value) unless otherwise specified. That means, you need to keep at least 5 significant digits (e.g., 12.34, 0.01234, etc.) in your final answer, and a couple of more digits in the intermediate steps.

Partial Fractions

1.  
   \(\displaystyle\int\frac{8}{x^2+12x+27}\,dx= \ln \left(\rule{0pt}{15pt}\right.\)\(\left.\rule{0pt}{15pt}\right) + C\)
   * Assume `x>0`.
     Hint: \(\log(AB) = \log A + \log B,~~ \log(A/B) = \log A - \log B,~~ k\log A = \log(A^k)\)
   
   
2.  
   \(\displaystyle\int\frac{5x+2}{x^2-12x+32}\,dx=\) \( + C\)
   * Use abs(x) for `|x|`.
   
   
3.  
   \(\displaystyle\int\frac{x^2+8x+64}{(x^2-16)(x+1)}\,dx=\) \( + C\)
   * Hint: Factor the denominator.
   
   
4.  
   \(\displaystyle\int\frac{7x^2+2x+45}{(x^2+5)(x-1)}\,dx=\) \( + C\)
   * Hint: The denominator is already fully factored.
   
   

   Improper Integrals

5.  
   \(\displaystyle\int_{0}^{\infty}x\, e^{-x^2/9}\,dx=\)
   
   
   
6.  
   \(\displaystyle\int_{1}^{\infty}\frac{dx}{x^6}=\)
   
   
   
7.  
   \(\displaystyle\int_{-\infty}^{\infty} \frac{dx}{x^2 + 16}\,=\)
   * Hint: Use substitution \(x=a\tan\theta\). Note that \(\tan(\pm\pi/2)=\pm\infty\).
   
   

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