Q: Which identity is used to reduce the power when integrating $$\sin^2$(x) or$$ $$\cos^2$(x)$$?

$$\cos^2$(\theta) = \frac{1 + \cos(2\theta)}{2}$$

Q: What is the best strategy for integrating $$\int \tan^3$(x)\sec^7$(x)\,dx$$?

Let $$u = \sec(x)$$, $$du = \sec(x)\tan(x)dx$$

Q: Evaluate $$\int \sec^4$(5x)\,dx.$$

$$\frac{1}{15}\tan^3$(5x) + \frac{1}{5}\tan(5x) + C$$

Q: Which of the following is the correct reduction formula for $$\int \sec^n(x)\,dx$$ when $$n is $$odd?

$$\int \sec^n(x)dx = \frac{\sec$$^{n-2}$$(x)\tan(x)}{n-1} + \frac{n-2}{n-1}\int \sec$$^{n-2}$$(x)dx$$

Q: Which substitution is most effective for $$\int \tan^4$(x)\sec^2$(x)\,dx$$?

Let $$u = \tan(x)$$, $$du = \sec^2$(x)dx$$

Q: Evaluate $$\int \sec^3$(x)\,dx$$ using integration by parts.

$$\frac{1}{2}\sec(x)\tan(x) + \frac{1}{2}\ln|\sec(x) + \tan(x)| + C$$

Question 1

Which substitution is most appropriate for evaluating $$\int \sin^5$(x) \cos$$^{18}$$(x)\,dx$$?

Accepted Answer

Let $$u = \cos(x)$$, $$du = -\sin(x)dx$$

Question 2

Evaluate $$\int \cos^2$(3x)\,dx.$$

Accepted Answer

$$\frac{x}{2} + \frac{\sin(6x)}{12} + C$$

Question 3

Which identity is used to reduce the power when integrating $$\sin^2$(x) or$$ $$\cos^2$(x)$$?

Accepted Answer

$$\cos^2$(	heta) = \frac{1 + \cos(2	heta)}{2}$$

Question 4

Evaluate $$\int \cos^2$(x)\sin^2$(x)\,dx.$$

Accepted Answer

$$\frac{1}{8}x - \frac{1}{32}\sin(4x) + C$$

Question 5

What is the best strategy for integrating $$\int 	an^3$(x)\sec^7$(x)\,dx$$?

Accepted Answer

Let $$u = \sec(x)$$, $$du = \sec(x)	an(x)dx$$

Question 6

Evaluate $$\int \sec^4$(5x)\,dx.$$

Accepted Answer

$$\frac{1}{15}	an^3$(5x) + \frac{1}{5}	an(5x) + C$$

Question 7

Which of the following is the correct reduction formula for $$\int \sec^n(x)\,dx$$ when $$n is $$odd?

Accepted Answer

$$\int \sec^n(x)dx = \frac{\sec$$^{n-2}$$(x)	an(x)}{n-1} + \frac{n-2}{n-1}\int \sec$$^{n-2}$$(x)dx$$

Question 8

Evaluate $$\int \sin^3$(x)\cos^2$(x)\,dx.$$

Accepted Answer

$$-\frac{1}{5}\cos^5$(x) + \frac{1}{3}\cos^3$(x) + C$$

Question 9

Which substitution is most effective for $$\int 	an^4$(x)\sec^2$(x)\,dx$$?

Accepted Answer

Let $$u = 	an(x)$$, $$du = \sec^2$(x)dx$$

Question 10

Evaluate $$\int 	an^4$(x)\sec^2$(x)\,dx.$$

Accepted Answer

$$\frac{1}{5}	an^5$(x) + C$$

Question 11

A medical device uses the signal $$f(x) = \sin^2$(x) to $$model a periodic pulse. What is the average value of $$f(x)$$ over one period $$[0, 2\pi]$$?

Accepted Answer

$$\frac{1}{2}$$

Question 12

Evaluate $$\int \sec^3$(x)\,dx$$ using integration by parts.

Accepted Answer

$$\frac{1}{2}\sec(x)	an(x) + \frac{1}{2}\ln|\sec(x) + 	an(x)| + C$$

Question 13

Which of the following is a correct strategy for integrating $$\int \sin^4$(x)\cos$$^{18}$$(x)\,dx$$?

Accepted Answer

Use the double angle identities to reduce the powers since both exponents are even.

Question 14

Evaluate $$\int 	an^2$(x)\sec^4$(x)\,dx.$$

Accepted Answer

$$\frac{1}{3}	an^3$(x) + 	an(x) + C$$

Question 15

A physicist models the intensity of a light wave as $$I(x) = \cos^2$(x). $$What is the total intensity over one period $$[0, 2\pi]$$?

Accepted Answer

$$\pi$$

Question 16

Which of the following is the correct substitution for $$\int \sin^2$(x)\cos^3$(x)\,dx$$?

Accepted Answer

Let $$u = \cos(x)$$, $$du = -\sin(x)dx$$

Question 17

Evaluate $$\int \sin^2$(x)\,dx.$$

Accepted Answer

$$\frac{x}{2} - \frac{1}{4}\sin(2x) + C$$

Question 18

A chemical reaction's rate is modeled by $$R(t) = \sin^2$(2t). $$What is the average rate over $$[0, \pi]$$?

Accepted Answer

$$\frac{1}{2}$$

Question 19

Which of the following is the correct integral for $$\int 	an^2$(x)\,dx$$?

Accepted Answer

$$	an(x) - x + C$$

Techniques of Integration: Trigonometric Integrals

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