Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 01

MATH101 A April 2024

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[hide] Question 01
Find $F'(x)$, given $${\begin{aligned}F(x)=\int _{0}^{x^{2}}{\frac {4-t}{1+\cos ^{2}(t)}}dt.\end{aligned}}$$

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!

[show] Hint
What does the fundamental theorem of calculus say? Note that the upper bound of the integral defining $F$ is $x^{2}$, not $x$.

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[show] Solution

One version of the fundamental theorem of calculus is $${\frac {\mathrm {d} }{\mathrm {d} x}}\int _{a}^{x}f(t)\mathrm {d} t=f(x).$$ In our case, $F$ is a composite $H\circ g$ of the two functions $${\begin{aligned}H(x)&=\int _{0}^{x}{\frac {4-t}{1+\cos ^{2}(t)}}\mathrm {d} t,\\g(x)&=x^{2}.\end{aligned}}$$ We wish to apply the chain rule. The derivatives are $${\begin{aligned}H'(x)&={\frac {4-x}{1+\cos ^{2}(x)}}\\g'(x)&=2x,\end{aligned}}$$ so we have $$F'(x)=H'(g(x))\cdot g'(x)={\frac {4-x^{2}}{1+\cos ^{2}(x^{2})}}\cdot 2x.$$

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