Question

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 solving$\int \frac{1}{x^2-4}dx$.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving$\int \frac{1}{\sqrt{x^2-4}}dx$.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving$\int \frac{1}{x^2+4}dx.$

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving$\int {\left(x^2+4\right)}^{-5/2}dx$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^2-a^2}$.

(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 $x=a\sin u$, 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

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

Step 1. Given information.

given,

Determine true or false, according to the question.

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

The statement is true.

Because $x=2\sec u$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 $\sqrt{x^2-a^2}$.

So, the given integral is $\int \frac{1}{x^2-4}dx$

Hence, the statement is true.

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 $\sqrt{x^2-a^2}$.

So, the given integral is $\int \frac{1}{\sqrt{x^2-4}}dx$

Hence, the statement is true.

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

The statement is true.

Because $=2\tan u$is in the form of $x=a\tan u$.

So, $a=2$.

When the trigonometric substitution is in the form of $x=a\tan u$, then this substitution can be used is in the integral form of $x^2+a^2$.

So, the given integral is $\int \frac{1}{x^2+4}dx$

Hence, the statement is true.

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

The statement is false.

Because $x=2\sin u$is in the form of $x=a\sin u$.

So, $a=2$.

When the trigonometric substitution is in the form of $x=a\sin u$, then this substitution can be used is in the integral form of $\sqrt{a^2-x^2}.$.

But, the given integral is $\int {\left(x^2+4\right)}^{-\frac{5}{2}}dx$

Therefore, the statement is false.

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

The statement is false.

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

The statement is true.

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

The statement is false.

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

The statement is true.