Science:Math Exam Resources/Courses/MATH103/April 2010/Question 01 (a)

MATH103 April 2010

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q2  •  Q3 (a)  •  Q3 (b)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  Q8  •  

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Question 01 (a)

Multiple Choice Question: Exactly ONE of the answers provided is correct. There is no partial credit in this questions. To which of the following integrals does the Fundamental Theorem of Calculus apply? i. $\int _{-1}^{1}{\frac {\cos x}{\ln x}}\,dx$ ii. $\int _{-1}^{1}{\frac {\cos x}{x}}\,dx$ iii. $\int _{-1}^{1}{\frac {\cos x}{{\sqrt {x}}+1}}\,dx$ iv. $\int _{-1}^{1}{\frac {\cos x}{x^{2}+1}}\,dx$ v. None of the above.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The Fundamental Theorem of Calculus only applies to functions that are continuous on the entire domain of integration.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Following the hint we check if the integrands above are continuous on the entire domain of integration. i. $\int _{-1}^{1}{\frac {\cos x}{\ln x}}\,dx.$ Since $\ln x$ is not even defined for negative numbers, the integrand ${\frac {\cos x}{\ln x}}$ is certainly not continuous on $(-1,1)$. ii. $\int _{-1}^{1}{\frac {\cos x}{x}}\,dx.$ The integrand is not defined for x=0, so it is certainly not continuous on $(-1,1)$. iii. $\int _{-1}^{1}{\frac {\cos x}{{\sqrt {x}}+1}}\,dx.$ Since ${\sqrt {x}}$ is not even defined for negative numbers, the integrand ${\frac {\cos x}{{\sqrt {x}}+1}}$ is certainly not continuous on $(-1,1)$. iv. $\int _{-1}^{1}{\frac {\cos x}{x^{2}+1}}\,dx.$ Here the integrand is defined everywhere. As a composition of continuous functions, the integrand is continuous. Hence the Fundamental Theorem of Calculus applies here. Final answer: iv.

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