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List some things which would suggest that a certain substitution \(u(x)\) could be a useful choice. What do you look for when choosing \(u(x)?\)

Short Answer

Expert verified

1. Derivative should be there.

2. Solve integral with basic formula.

Step by step solution

01

Introduction

Most integral solve by using substitution. By letting function to some other variable.

02

Explanation

In substitution choose function as \(u(x)\).

List of things that need to be considered while substitution:-

1. Derivative of \(u(x)\) should be in integrand.

2. After substitution integral should be easy or use basic formula.

3. After integration need to replace with the original variable.

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Most popular questions from this chapter

Solve given definite integrals.

โˆซ04xx2+4dx

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solvingโˆซ1x2โˆ’4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solvingโˆซ1x2โˆ’4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solvingโˆซ1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingโˆซx2+4โˆ’5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2โˆ’a2.

(f) True or False: Trigonometric substitution doesnโ€™t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

Find three integrals in Exercises 39โ€“74 that can be solved without using trigonometric substitution.

Complete the square for each quadratic in Exercises 28โ€“33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x2-5x+1

Solve โˆซx+2x2+4x32dxthe following two ways:

(a) with the substitution u=x2+4x;

(b) by completing the square and then applying the trigonometric substitution x + 2 = 2 sec u.

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