Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 20 (a)

MATH101 B April 2024

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Question 20 (a)
Let $f(x)=\int _{0}^{x}te^{-t}\;\mathrm {d} t.$ Use this function in all parts below. (a) Solve the integral to produce a simple formula for $f(x).$

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
Consider breaking the integrand into parts where one function simplifies upon differentiation, while the other remains easy to integrate. Can you think of one such integration method?

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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. We can find a simple for $f(x)$ by integration by parts. Let $u=t,\;du=dt$ and $dv=e^{-t}dt,\;v=-e^{-t}.$ Then we have $${\begin{aligned}f(x)&=\int _{0}^{x}te^{-t}dt\\&=\left[t(-e^{-t})\right]_{0}^{x}-\int _{0}^{x}(-e^{-t})dt\\&=-xe^{-x}+\left[-e^{-t}\right]_{0}^{x}\\&=-xe^{-x}-\left[e^{-x}-1\right]\\&=1-(x+1)e^{-x}\end{aligned}}$$