Science:Math Exam Resources/Courses/MATH101/April 2018/Question 01 (iii)

MATH101 April 2018

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Question 01 (iii)
Evaluate ${\textstyle \displaystyle \int _{-\pi }^{\pi }\left[1+x^{4}\sin(x^{3})\right]\,dx}$ . Simplify your answer completely.

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
Observe that ${\textstyle x^{4}\sin(x^{3})}$ is an odd function and the integral is over a symmetric interval centered at zero.

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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. The trick here is to notice that ${\textstyle x^{4}\sin(x^{3})}$ is an odd function (since ${\textstyle x^{4}}$ is an even function and ${\textstyle sin(x^{3})}$ is an odd function) and the integral is over a symmetric interval which is centered at zero. Recall that integrating an odd function over such an interval always gives the value zero, so after applying linearity of the integral this complicated term will disappear and we get $\int _{-\pi }^{\pi }{\big [}1+x^{4}\sin(x^{3}){\big ]}\;dx=\int _{-\pi }^{\pi }1dx+\int _{-\pi }^{\pi }x^{4}\sin(x^{3})\;dx=\int _{-\pi }^{\pi }1dx=2\pi .$ Answer: The correct answer is ${\textstyle {\color {blue}2\pi }}$ .