Science:Math Exam Resources/Courses/MATH103/April 2011/Question 02 (a)
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Question 02 (a) 

Evaluate the integral 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Use a trigonometric identity to rewrite the integrand. 
Hint 2 

. 
Hint 3 

Alternative solution: Use a substitution. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. We first solve the indefinite integral, using the trigonometric identity :
Therefore Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} I_1 &= \int_{1/2}^1 \sin(\pi x)\cos(\pi x)\,dx \\ &= \left.\frac{1}{4\pi}\cos(2\pi x)\right_{1/2}^1 \\ &= \frac{1}{4\pi} \cos(2\pi) + \frac1{4\pi} \cos(\pi) \\ &= \frac1{4\pi}  \frac1{4\pi} = \frac1{2\pi}. \end{align} } 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Alternatively, substitute Then , when then , and when then Therefore
