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MAT136H5 S 2025 - All Sections 1.3 Preparation Check

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Please read the Fundamental Theorem of Calculus Part 2 (Theorem 1.5).    In this problem we work through the steps of evaluating the definite integral ∫ 9 1 5 4 √ x3 dx. a) Find an antiderivative of  f(x)= 5 4 √ x3 . (Re-read section 4.10 from MAT135 Links to an external site.  if you are unsure.) An antiderivative of f is: F(x)=  [ Select ] -(5/2) x^(-1/2) -(5/4) x^(-1/2) -(1/2) x^(-5/2) -(5/2) x^(-5/2) b) What is the final answer? The definite integral ∫ 9 1 5 4 √ x3 dx = [ Select ] 3/5 5/4 (5/4)^3 - 1 5/3 1 - (5/4)^3 -5/4 -5/3   (Use a simple calculator for (b) if you want.)

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Step into the problem by parsing the integrand. The expression suggests f(x) = (5/4) x^(-3/2) because the antiderivative options involve x^(-1/2) terms upon integrating x^(-3/2). Now analyze each option for part (a): - Option 1: -(5/2) x^(-1/2). If you differentiate this, you get -(5/2) * (-1/2) x^(-3/2) = (5/4) x^(-3/2), which matches f(x). This makes it a correct antiderivative. - Option 2: -(5/4) x^(-1/2). Differentiating yields -(5/4) * (......Login to view full explanation

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