$$\tan(x) - \int \tan^2$(x) \tan(x) \, dx = \tan(x) - \int (\sec^2$(x) - 1) \tan(x) \, dx = \frac{1}{2} \tan^2$(x) + \ln|\sec x| + C$$

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

Question 1

Which substitution is most appropriate for evaluating the integral $$\int \frac{dx}{\sqrt{9 - x^2$}}$$?

Accepted Answer

Let $$x = 3 \sin 	heta$$

Question 2

Evaluate $$\int \cos^2$(3x) \, dx$$ using a suitable trigonometric identity.

Accepted Answer

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

Question 3

Which identity is used to reduce the power of $$\sin^2$(\alpha) in $$integrals?

Accepted Answer

$$\sin^2$(\alpha) = \frac{1 - \cos(2\alpha)}{2}$$

Question 4

For the integral $$\int \sin^5$(x) \cos^2$(x) \, dx$$, which substitution is most effective?

Accepted Answer

Let $$u = \cos x$$

Question 5

What is the area enclosed by the ellipse $$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} = 1$$?

Accepted Answer

$$\pi ab$$

Question 6

Which trigonometric substitution is appropriate for $$\int \frac{dx}{x^2$ + a^2$}$$?

Accepted Answer

Let $$x = a 	an 	heta$$

Question 7

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

Accepted Answer

$$	an(x) - \int 	an^2$(x) 	an(x) \, dx = 	an(x) - \int (\sec^2$(x) - 1) 	an(x) \, dx = \frac{1}{2} 	an^2$(x) + \ln|\sec x| + C$$

Question 8

Which substitution is best for evaluating $$\int \frac{dx}{\sqrt{x^2$ - a^2$}}$$?

Accepted Answer

Let $$x = a \sec 	heta$$

Question 9

What is the result of $$\int \sin(4x) \cos(5x) \, dx$$?

Accepted Answer

$$\frac{1}{2} (\sin(9x) - \sin(x)) + C$$

Question 10

Which identity is used to reduce the power of $$\cos^2$(\alpha) in $$integrals?

Accepted Answer

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

Question 11

For the integral $$\int 	an^6$(x) \sec^4$(x) \, dx$$, which substitution is most effective?

Accepted Answer

Let $$u = 	an x$$

Question 12

What is the result of $$\int \sec^3$(x) \, dx$$?

Accepted Answer

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

Question 13

Which substitution is appropriate for evaluating $$\int \frac{dx}{\sqrt{x^2$ + a^2$}}$$?

Accepted Answer

Let $$x = a 	an 	heta$$

Question 14

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

Accepted Answer

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

Question 15

Which trigonometric substitution is appropriate for $$\int \frac{dx}{\sqrt{a^2$ - x^2$}}$$?

Accepted Answer

Let $$x = a \sin 	heta$$

Question 16

What is the result of $$\int \sin^4$(x) \, dx$$?

Accepted Answer

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

Question 17

Which substitution is best for evaluating $$\int \frac{dx}{x^2$ - a^2$}$$?

Accepted Answer

Let $$x = a \sec 	heta$$

Question 18

What is the result of $$\int \cos^2$(x) \, dx$$?

Accepted Answer

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

Trigonometric Integrals and Trigonometric Substitution

스터디 가이드 - 연습 문제