$$\frac{1}{2}\sec x \tan x + \frac{1}{2}\int \sec x \, dx$$

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

$$\int \tan^n x \, dx = \frac{1}{n-1}\tan$$^{n-1}$$ x - \int \tan$$^{n-2}$$ x \, dx$$

Question 1

Which substitution is most appropriate for evaluating $$\int \sin^3$ x \cos^2$ x \, dx$$?

Accepted Answer

Let $$u = \cos x$$, so $$du = -\sin x \, dx$$ and rewrite $$\sin^3$ x as$$ $$(1-\cos^2$ x)\sin x$$

Question 2

Evaluate $$\int \sin^2$ x \cos^2$ x \, dx$$ using a reduction formula.

Accepted Answer

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

Question 3

Which formula is used to reduce the power of $$\sin^n x$$ when $$n is $$even?

Accepted Answer

$$\sin^2$ x = \frac{1 - \cos 2x}{2}$$

Question 4

What is the result of $$\int \sec^3$ x \, dx$$ using integration by parts?

Accepted Answer

$$\frac{1}{2}\sec x 	an x + \frac{1}{2}\int \sec x \, dx$$

Question 5

Which substitution is recommended for $$\int 	an^n x \, dx$$ when $$n is $$odd?

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Which product-to-sum formula is used to simplify $$\sin mx \cos nx$$?

Accepted Answer

$$\sin mx \cos nx = \frac{1}{2}[\sin(m+n)x + \sin(m-n)x]$$

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

$$\int 	an^n x \, dx = \frac{1}{n-1}	an$$^{n-1}$$ x - \int 	an$$^{n-2}$$ x \, dx$$

Question 10

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

Accepted Answer

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

Question 11

Which technique is most effective for integrating $$\int \sec^2$ x 	an x \, dx$$?

Accepted Answer

Substitute $$u = 	an x$$

Question 12

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

Accepted Answer

$$\ln|\sec x + 	an x| + C$$

Question 13

Which of the following is a valid product-to-sum formula for $$\cos mx \cos nx$$?

Accepted Answer

$$\cos mx \cos nx = \frac{1}{2}[\cos(m-n)x + \cos(m+n)x]$$

Question 14

Evaluate $$\int \sin^4$ x \, dx.$$

Accepted Answer

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

Question 15

Which substitution is most useful for $$\int \sin^n x \cos^m x \, dx$$ when $$n is $$odd?

Accepted Answer

Let $$u = \cos x$$

8.2 Trigonometric Integrals

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