Q: Which identity is useful when making the substitution $$x = a \sin \theta in $$the integral $$\int \sqrt{a^2$ - x^2$} \, dx$$?

$$a^2$$ - $$x^2$$ = $$a^2$$ \$$cos^2$$ \theta$$

Q: Which of the following is the correct range for $$\theta$$ when using the substitution $$x = a \tan \theta$$?

$$-\frac{\pi}{2} \u003c \theta \u003c \frac{\pi}{2}$$

Question 1

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

Accepted Answer

$$x = a \sin 	heta$$

Question 2

Given the integral $$\$$int_0$$^a \sqrt{a^2$ - x^2$} \, dx$$, what real-world quantity does this integral represent?

Accepted Answer

The area of a semicircle of radius $$a$$

Question 3

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

Accepted Answer

$$a \sec 	heta$$

Question 4

Evaluate $$\int \frac{dx}{\sqrt{x^2$ + 4}}.$$

Accepted Answer

$$\ln \left| x + \sqrt{x^2$ + 4} \right| + C$$

Question 5

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

Accepted Answer

$$x = a \sec 	heta$$

Question 6

Suppose you are calculating the arc length of the parabola $$y = x^2$ from x = 0$ to x = 1$. Which type of substitution is likely to be useful?

Accepted Answer

Tangent substitution

Question 7

Given $$x = a \sin 	heta$$, what is $$dx in $$terms of $$	heta$$?

Accepted Answer

$$a \cos 	heta \, d	heta$$

Question 8

Which of the following integrals would be best approached using the substitution $$x = a 	an 	heta$$?

Accepted Answer

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

Question 9

If $$x = a \sec 	heta$$, what is $$\sqrt{x^2$ - a^2$} in $$terms of $$	heta$$?

Accepted Answer

$$a 	an 	heta$$

Question 10

A physicist is modeling the gravitational field around a circular disk. The integral $$\int_{-a}^a \sqrt{a^2$ - x^2$} \, dx$$ appears in their calculations. What does this integral represent?

Accepted Answer

The area of a full circle of radius $$a$$

Question 11

Which identity is useful when making the substitution $$x = a \sin 	heta in $$the integral $$\int \sqrt{a^2$ - x^2$} \, dx$$?

Accepted Answer

$$a^2$$ - $$x^2$$ = $$a^2$$ \$$cos^2$$ 	heta$$

Question 12

Evaluate $$\int \sqrt{4 - x^2$} \, dx$$ using an appropriate trigonometric substitution.

Accepted Answer

$$\frac{x}{2} \sqrt{4 - x^2$} + 2 \arcsin\left(\frac{x}{2}\right) + C$$

Question 13

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

Accepted Answer

$$x = a \sin 	heta$$

Question 14

A medical imaging device uses the formula $$\$$int_0$$^a \sqrt{a^2$ - x^2$} \, dx to $$calculate the cross-sectional area of a circular artery. If $$a = 3 mm$$, what is the area?

Accepted Answer

$$\frac{9\pi}{2} mm$^2$$

Question 15

Which of the following is the correct range for $$	heta$$ when using the substitution $$x = a 	an 	heta$$?

Accepted Answer

$$-\frac{\pi}{2} \u003c 	heta \u003c \frac{\pi}{2}$$

Question 16

Evaluate $$\int \frac{dx}{\sqrt{x^2$ - 9}}.$$

Accepted Answer

$$\ln \left| x + \sqrt{x^2$ - 9} \right| + C$$

Trigonometric Substitutions in Integration

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