Q: What is the range of $$\theta$$ for the substitution $$x = a \sin \theta$$?

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

Q: After substituting $$x = a \tan \theta in$$ $$\int \frac{dx}{\sqrt{x^2$ + a^2$}}$$, what does $$dx$$ become?

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

Question 1

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

Accepted Answer

$$x = a \sin 	heta$$

Question 2

Given the integral $$\int \frac{dx}{\sqrt{x^2$ + 9}}$$, which substitution will simplify the integrand?

Accepted Answer

$$x = 3 	an 	heta$$

Question 3

After making the substitution $$x = a \sin 	heta in$$ $$\int \sqrt{a^2$ - x^2$} \, dx$$, what does $$\sqrt{a^2$ - x^2$}$$ become?

Accepted Answer

$$a \cos 	heta$$

Question 4

What is the range of $$	heta$$ for the substitution $$x = a \sin 	heta$$?

Accepted Answer

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

Question 5

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 6

Suppose you are evaluating $$\int \frac{dx}{\sqrt{16 - x^2$}}. $$What is the result after using the appropriate trigonometric substitution?

Accepted Answer

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

Question 7

If $$x = a 	an 	heta$$, what does $$\sqrt{x^2$ + a^2$}$$ simplify to?

Accepted Answer

$$a \sec 	heta$$

Question 8

A ladder of length $$10 ft is $$leaning against a wall. If the distance from the wall to the base is $$x ft$$, which trigonometric substitution would you use to find the height of the ladder on the wall?

Accepted Answer

$$x = 10 \sin 	heta$$

Question 9

Which of the following is the correct substitution for $$\int \frac{dx}{x^2$ \sqrt{x^2$ - 25}}$$?

Accepted Answer

$$x = 5 \sec 	heta$$

Question 10

After substituting $$x = a 	an 	heta in$$ $$\int \frac{dx}{\sqrt{x^2$ + a^2$}}$$, what does $$dx$$ become?

Accepted Answer

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

Question 11

Evaluate $$\int \sqrt{9 - x^2$} \, dx$$ using trigonometric substitution.

Accepted Answer

$$\frac{x}{2} \sqrt{9 - x^2$} + \frac{9}{2} \arcsin \left( \frac{x}{3} \right) + C$$

Question 12

Which triangle would you draw to represent the substitution $$x = a 	an 	heta$$?

Accepted Answer

A right triangle with opposite side $$x$$, adjacent side $$a$$, and hypotenuse $$\sqrt{x^2$ + a^2$}$$

Question 13

What is the main purpose of using trigonometric substitutions in integrals involving square roots?

Accepted Answer

To eliminate the square root and simplify the integral

Question 14

If $$x = a \sec 	heta$$, what does $$\sqrt{x^2$ - a^2$}$$ become?

Accepted Answer

$$a 	an 	heta$$

Question 15

Which of the following is a correct result for $$\int \frac{dx}{\sqrt{x^2$ - a^2$}}$$?

Accepted Answer

$$\ln \left| x + \sqrt{x^2$ - a^2$} \right| + C$$

Trig Substitutions in Integration

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