Question 1

Which substitution is most appropriate when integrating $$\int \sin^m x \cos^n x \, dx$$ and $$m is $$odd?

Accepted Answer

Let $$u = \sin x$$, then $$du = \cos x \, dx$$

Question 2

Evaluate $$\int \sec^2$ x \, dx.$$

Accepted Answer

$$	an x + C$$

Question 3

When integrating $$\int \sin^m x \cos^n x \, dx$$, if both $$m$$ and $$n$$ are even, which identity is most useful?

Accepted Answer

Double angle identities: $$\sin^2$ x = \frac{1 - \cos 2x}{2}$$ and $$\cos^2$ x = \frac{1 + \cos 2x}{2}$$

Question 4

Which of the following integrals can be solved directly using a basic trigonometric identity?

Accepted Answer

$$\int \sin x \cos x \, dx$$

Question 5

What is the result of $$\int \sin^2$ x \, dx$$?

Accepted Answer

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

Question 6

A medical device uses the function $$f(x) = \sin^3$ x \cos^2$ x to $$model a periodic signal. Which substitution is most effective for integrating this function?

Accepted Answer

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

Question 7

Which integral is equivalent to $$\int \sec^2$ x \, dx$$?

Accepted Answer

$$\int \frac{1}{\cos^2$ x} \, dx$$

Question 8

A chemical reaction rate is modeled by $$R(t) = \sin^2$ t \cos^2$ t. $$What is $$\int \sin^2$ t \cos^2$ t \, dt$$?

Accepted Answer

$$\frac{1}{8} \sin 4t + \frac{t}{4} + C$$

Question 9

Which of the following is the correct reduction formula for $$\int \sin^n x \, dx$$?

Accepted Answer

$$-\frac{1}{n} \sin$$^{n-1}$$ x \cos x + \frac{n-1}{n} \int \sin$$^{n-2}$$ x \, dx$$

Question 10

Evaluate $$\int \sin^3$ x \, dx.$$

Accepted Answer

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

Question 11

A sound wave is modeled by $$f(x) = \sin^2$ x. $$What is the average value of $$f(x)$$ over $$[0, \pi]$$?

Accepted Answer

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

Question 12

Which of the following is a valid strategy for integrating $$\int \sin^m x \cos^n x \, dx$$ when $$n is $$odd?

Accepted Answer

Save one $$\cos x$$ factor and use $$\cos^2$ x = 1 - \sin^2$ x$$

Question 13

Evaluate $$\int \sin x \cos x \, dx.$$

Accepted Answer

$$\frac{1}{2} \sin^2$ x + C$$

Question 14

Which trigonometric identity is most useful for integrating $$\int \sin x \cos x \, dx$$?

Accepted Answer

$$\sin 2x = 2 \sin x \cos x$$

Question 15

A historical event is modeled by $$f(t) = \cos^4$ t. $$What is $$\int \cos^4$ t \, dt$$?

Accepted Answer

$$\frac{3}{8} t + \frac{1}{4} \sin 2t + \frac{1}{32} \sin 4t + C$$

Question 16

Which of the following is the correct reduction formula for $$\int \cos^n x \, dx$$?

Accepted Answer

$$\frac{1}{n} \cos$$^{n-1}$$ x \sin x + \frac{n-1}{n} \int \cos$$^{n-2}$$ x \, dx$$

Question 17

Evaluate $$\int \sin^2$ x \cos^2$ x \, dx$$ using a double angle identity.

Accepted Answer

$$\frac{1}{8} \sin 4x + \frac{x}{4} + C$$

Question 18

A periodic signal in engineering is modeled by $$f(x) = \sin^2$ x \cos^2$ x. $$What is the average value of $$f(x)$$ over $$[0, 2\pi]$$?

Accepted Answer

$$\frac{1}{4}$$

Integration of Products of Powers of Sines and Cosines

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