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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 solving1x24dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingx2+45/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2a2.

(f) True or False: Trigonometric substitution doesn’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.

Short Answer

Expert verified

Ans:

(a) Statement is True.

(b) Statement is True.

(c) Statement is True.

(d) Statement is False.

(e) Statement is False.

(f) Statement is True.

(g) Statement is False.

(h) Statement is True.

Step by step solution

01

Step 1. Given information.

given,

Determine true or false, according to the question.

02

Step 2. (a)  The objective is to determine the given statement is true or false.

The statement is true.

Because x=2secuis in the form of x=asecu.

So, a=2.

When the trigonometric substitution is in the form of x=asecu, then this substitution can be used is in the integral form of x2a2.

So, the given integral is 1x24dx

Hence, the statement is true.

03

Step 3. (b)  The objective is to determine the given statement is true or false. 

The statement is true.

Because x=2sec⁡u is in the form of x=asecu.

So, a=2.

When the trigonometric substitution is in the form of x=asecu, then this substitution can be used is in the integral form of x2a2.

So, the given integral is 1x24dx

Hence, the statement is true.

04

Step 4. (c)  The objective is to determine the given statement is true or false.  

The statement is true.

Because =2tanuis in the form of x=atanu.

So, a=2.

When the trigonometric substitution is in the form of x=atanu, then this substitution can be used is in the integral form of x2+a2.

So, the given integral is 1x2+4dx

Hence, the statement is true.

05

Step 5. (d)  The objective is to determine the given statement is true or false. 

The statement is false.

Because x=2sinuis in the form of x=asinu.

So, a=2.

When the trigonometric substitution is in the form of x=asinu, then this substitution can be used is in the integral form of a2x2..

But, the given integral is x2+452dx

Therefore, the statement is false.

06

Step 6. (e)   The objective is to determine the given statement is true or false.  

The statement is false.

07

Step 7.  (f)  The objective is to determine whether the given statement is true or false.  

The statement is true.

08

Step 8. (g)  The objective is to determine whether the given statement is true or false.  

The statement is false.

09

Step 9. (h)  The objective is to determine whether the given statement is true or false.  

The statement is true.

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