Proposed exercises from chapter 8.1 of Larson and Hostetler's calculus book. Volume I: Selecting the correct antiderivative
In the proposed exercises of the book of calculus with analytical geometry by Larson and Hostetler in chapter 8 whose theme is: Integration techniques, L'hopital's rule and improper integrals, there are a series of exercises ranging from 1 to 4, in which you must select the correct antiderivative, the purpose of this post is to perform the exercise number 1 and choose among the four options the correct one.
Exercise 1 of chapter 8. section 8.1
Given the following differential equation:
Select the correct anti-derivative:
The solution is to solve the differential equation by clearing the differential of x (dx) and applying the integration process on both sides of the equality, as follows:
On the left side of the equality, the integral of dy is equal to dy, so we would be left with the following:
It only remains to solve the integral that is on the right side of the equality, for this the method of substitution or change of variable is applied, as follows:
Once we have already made the change of variable, where we are calling U=x2+1 and when we clear xdx = 1/2du, we solve the integral in terms of the variable U, applying the powering rule for integrals:
Once we have already applied the power rule for integrals, we simplify the fractions and return the change of variable, where the U is we place x2+1:
As a lesson learned, we have that the integration technique that best applies to solve the integral proposed in chapter 8 of Larson's book Volume I of section 8.1 is the technique of substitution or change of variable.
The correct answer is option b.)
Bibliographic Reference
Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. Año 2006