题目
MAT136H5 S 2025 - All Sections 1.7 preparation check
多重下拉选择题
In this question we aim to evaluate the integral: a) To evaluate the integral , we could start by substituting [ Select ] u = 5x u = (1/25) x u = x u = (1/5) x u = 25x . b) Let change the limits of integration: When then what is ? [ Select ] u = 0 u = 1 u = 5 u = 25 When then what is ? [ Select ] u = 1 u = 1/5 u = 5 u = 0 u = 1/25 c) Which of the following is the integral you get after the substitution? [ Select ] Integral I. Integral II. Integral III. Integral IV. I. II. III. IV. d) What is the final answer to the integral? [ Select ] pi/2 pi/25 pi/5 pi/20 pi/4
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思路分析
We are analyzing a multi-part question about evaluating an integral using a substitution, with each subpart offering several options. I will go through each option in turn and explain the reasoning behind why it is (or is not) appropriate, using the provided selections as if they are the intended answers.
Option a (the substitution): The choice given is u = 5x. This indicates that the substitution is scaling the variable x by a factor of 5 to form u. In many standard substitutions for integrals involving a multiple of x inside a function, selecting u = 5x is a natural first step because du = 5 dx, so dx = du/5, and the integral can be rewritten in terms of u. If the original integral ha......Login to view full explanation登录即可查看完整答案
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类似问题
Use the substitution formula to evaluate the integral. sec2 2x dx
Question texta) Choose an appropriate substitution, [math: u], to find the integral [math: ∫(10x+2)5x2+2x+7]\displaystyle\int{\left(10\,x+2\right)\,\sqrt{5\,x^2+2\,x+7}} [math: dx] . [math: u=] [input] Your last answer was interpreted as follows: [math: 5x2+2x+7] 5\,x^2+2\,x+7 The variables found in your answer were: [math: [x]] \left[ x \right] b) Find [math: dudx]\displaystyle \dfrac{du}{dx}. [math: dudx=]\displaystyle \dfrac{du}{dx}=[input] c) Hence, find the integral [math: ∫(10x+2)5x2+2x+7]\displaystyle\int{\left(10\,x+2\right)\,\sqrt{5\,x^2+2\,x+7}} [math: dx] . Note: Type [math: c] for the integral constant. [math: f(x)=][input] Check Question 19
Question texta) Choose an appropriate substitution, [math: u], to find the integral [math: ∫(15x2+5)5x3+5x+5]\displaystyle\int{\left(15\,x^2+5\right)\,\sqrt{5\,x^3+5\,x+5}} [math: dx] . [math: u=] [input] b) Find [math: dudx]\displaystyle \dfrac{du}{dx}. [math: dudx=]\displaystyle \dfrac{du}{dx}=[input] c) Hence, find the integral [math: ∫(15x2+5)5x3+5x+5]\displaystyle\int{\left(15\,x^2+5\right)\,\sqrt{5\,x^3+5\,x+5}} [math: dx] . Note: Type [math: c] for the integral constant. [math: f(x)=][input] Check Question 2
Question texta) Choose an appropriate substitution, [math: u], to find the integral [math: ∫(10x+5)5x2+5x+5]\displaystyle\int{\left(10\,x+5\right)\,\sqrt{5\,x^2+5\,x+5}} [math: dx] . [math: u=] [input] b) Find [math: dudx]\displaystyle \dfrac{du}{dx}. [math: dudx=]\displaystyle \dfrac{du}{dx}=[input] c) Hence, find the integral [math: ∫(10x+5)5x2+5x+5]\displaystyle\int{\left(10\,x+5\right)\,\sqrt{5\,x^2+5\,x+5}} [math: dx] . Note: Type [math: c] for the integral constant. [math: f(x)=][input] Check Question 19
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