MATH3503 Online Assignment 7

• 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.

Trigonometric Integration

Integration using trigonometric identities

1.  
   \(\displaystyle\int_{-\pi/4}^{\pi/4} \cos^3(3x)\,dx=\)
   
   
   
2.  
   \(\displaystyle\int_{0}^{0.6} \sin^4(3x)\,dx=\)
   
   
   
3.  
   \(\displaystyle\int_{0}^{0.6} \sin^2(2x)\cos^3(2x)\,dx=\)
   
   
   
4.  
   \(\displaystyle\int_{-0.3}^{0} \sin^3(5x)\cos^{18}(5x)\,dx=\)
   
   
   
5.  
   \(\displaystyle\int_{0}^{0.6} \sin^{6}(3x)\,dx=\)
   * Use the reduction formula and the result of Question 2.
   
   

   Trigonometric Substitution

6.  
   \(\displaystyle\int_{0}^{8/5} \sqrt{64-25x^2\ }\,dx=\)
   
   
   
7.  
   \(\displaystyle\int_{0}^{9} \frac{x^3\,dx}{\sqrt{81-x^2\,}}\,=\)
   
   
   
8.  
   \(\displaystyle\int_{-8}^{8} \frac{dx}{\sqrt{x^2 + 4\,}}\,=\)
   * Hint: \(\displaystyle\int \sec\theta d\theta = \ln |\sec\theta+\tan\theta|+C\), and \(\sec^{-1}x = \cos^{-1}(\frac{1}{x})\).
   
   
9.  
   \(\displaystyle\int_{2}^{4} \frac{dx}{x\sqrt{x^2 - 4\,}}\,=\)
   
   
   
10.  
    \(\displaystyle\int_{-2}^{0} \frac{dx}{\sqrt{x^2 + 4x+5\,}}\,=\)
    * Hint: Complete the square, i.e., \(x^2+ax+b = (x+\frac{a}{2})^2 + b - \frac{a^2}{4}\).
    
    

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