Q: Which range for $$\theta$$ ensures that $$x = a \sin \theta is $$one-to-one for $$\sqrt{a^2$ - x^2$}$$?

$$\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

Q: Which of the following is a correct back-substitution for $$\theta if$$ $$x = 5 \sin \theta$$?

$$\theta = \arcsin \left( \frac{x}{5} \right)$$

Q: Why can absolute values sometimes be removed after trigonometric substitution?

Because the range of $$\theta$$ ensures the trigonometric function is always positive.

Q: If $$x = a \tan \theta$$, what is $$dx in $$terms of $$d\theta$$?

$$a \sec^2$ \theta \, d\theta$$

Q: Which trigonometric identity justifies the substitution $$x = a \sin \theta$$ for $$\sqrt{a^2$ - x^2$}$$?

$$\sin^2$ \theta + \cos^2$ \theta = 1$$

Question 1

Which trigonometric substitution is most appropriate for evaluating the integral $$\int \sqrt{9 - x^2$} \, dx$$?

Accepted Answer

$$x = 3 \sin 	heta$$

Question 2

Given the integral $$\int \frac{dx}{\sqrt{x^2$ + 16}}$$, which substitution will simplify the square root?

Accepted Answer

$$x = 4 	an 	heta$$

Question 3

For the integral $$\int \sqrt{x^2$ - 25} \, dx$$, what is the correct trigonometric substitution?

Accepted Answer

$$x = 5 \sec 	heta$$

Question 4

After making the substitution $$x = a \sin 	heta$$ for $$\sqrt{a^2$ - x^2$}$$, what does the square root simplify to?

Accepted Answer

$$a \cos 	heta$$

Question 5

What is the geometric interpretation of the substitution $$x = a 	an 	heta in $$the context of $$\sqrt{a^2$ + x^2$}$$?

Accepted Answer

It represents the tangent of the angle where the opposite side is $$x$$ and the adjacent side is $$a.$$

Question 6

Which of the following integrals is best solved using the substitution $$x = a \sec 	heta$$?

Accepted Answer

$$\int \sqrt{x^2$ - a^2$} \, dx$$

Question 7

If $$x = 2 \sin 	heta$$, what is $$dx in $$terms of $$d	heta$$?

Accepted Answer

$$2 \cos 	heta \, d	heta$$

Question 8

Evaluate $$\int \frac{dx}{\sqrt{16 - x^2$}}$$ using an appropriate trigonometric substitution.

Accepted Answer

$$\arcsin \left( \frac{x}{4} \right) + C$$

Question 9

Which range for $$	heta$$ ensures that $$x = a \sin 	heta is $$one-to-one for $$\sqrt{a^2$ - x^2$}$$?

Accepted Answer

$$	heta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}ight]$$

Question 10

After substituting $$x = 3 	an 	heta$$ into $$\sqrt{9 + x^2$}$$, what does the square root become?

Accepted Answer

$$3 \sec 	heta$$

Question 11

A medical imaging device uses the integral $$\int \frac{dx}{\sqrt{x^2$ - 36}} to $$model a signal. Which substitution should be used?

Accepted Answer

$$x = 6 \sec 	heta$$

Question 12

Which of the following is a correct back-substitution for $$	heta if$$ $$x = 5 \sin 	heta$$?

Accepted Answer

$$	heta = \arcsin \left( \frac{x}{5} ight)$$

Question 13

Why can absolute values sometimes be removed after trigonometric substitution?

Accepted Answer

Because the range of $$	heta$$ ensures the trigonometric function is always positive.

Question 14

Which substitution would you use to evaluate $$\int \sqrt{25 - x^2$} \, dx$$?

Accepted Answer

$$x = 5 \sin 	heta$$

Question 15

If $$x = a 	an 	heta$$, what is $$dx in $$terms of $$d	heta$$?

Accepted Answer

$$a \sec^2$ 	heta \, d	heta$$

Question 16

A physicist is modeling a pendulum and encounters $$\int \frac{dx}{\sqrt{49 - x^2$}}. $$What is the antiderivative?

Accepted Answer

$$\arcsin \left( \frac{x}{7} \right) + C$$

Question 17

Which trigonometric identity justifies the substitution $$x = a \sin 	heta$$ for $$\sqrt{a^2$ - x^2$}$$?

Accepted Answer

$$\sin^2$ 	heta + \cos^2$ 	heta = 1$$

Question 18

After substituting $$x = 4 \sec 	heta$$ into $$\sqrt{x^2$ - 16}$$, what does the square root become?

Accepted Answer

$$4 	an 	heta$$

Question 19

Which of the following is NOT a standard form for trigonometric substitution?

Accepted Answer

$$\sqrt{a^2$ + x}$$

Question 20

A civil engineer needs to evaluate $$\int \sqrt{100 + x^2$} \, dx$$ for a bridge cable. Which substitution should be used?

Accepted Answer

$$x = 10 	an 	heta$$

Trig Substitutions in Calculus: Techniques and Applications

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