Science:Math Exam Resources/Courses/MATH101/April 2013/Question 03 (a)

MATH101 April 2013

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[hide] Question 03 (a)
Short-Answer Questions. Question 1-3 are short-answer questions. Put your answer in the box provided. Simplify your answer as much as possible. Full Marks will be awarded for a correct answer placed in the box. Show your work, for part marks. Evaluate $\int _{1}^{3}(2x-1)e^{x^{2}-x}\,dx$

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.

[show] Hint 1
Always try a simple integration first, possibly with a nice substitution.

[show] Hint 2
Don't forget to change the bounds of integration!

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[show] Solution
Using substitution, ${\begin{aligned}u&=x^{2}-x\\du&=(2x-1)\,dx\end{aligned}}$ and the bounds change ${\begin{aligned}x=1&\Rightarrow u=0\\x=3&\Rightarrow u=6\end{aligned}}$ Then ${\begin{aligned}\int _{1}^{3}(2x-1)e^{x^{2}-x}\,dx&=\int _{0}^{6}e^{u}\,du\\&=e^{u}{\big \|}_{0}^{6}\\&=e^{6}-1\end{aligned}}$