Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

문제 8.3.34

8장, 문제 8.3.34

9–61. Trigonometric integrals Evaluate the following integrals.
34. ∫ tan⁹x sec⁴x dx

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Step 1: Recognize that the integral involves powers of tangent and secant. For trigonometric integrals involving these functions, it is often helpful to use trigonometric identities to simplify the expression. Recall the identity:

sec²x = 1 + tan²x

Step 2: Split the powers of secant and tangent to facilitate substitution. Specifically, reserve one

sec²x

term for substitution, as

d(tanx) = sec²x dx

. Rewrite the integral as:

∫ tan⁹x sec²x sec²x dx

Step 3: Perform substitution. Let

u = tanx

, which implies

du = sec²x dx

. Replace

tanx

with

u

and

sec²x dx

with

du

. The integral becomes:

∫ u⁹ sec²x du

Step 4: Simplify further using the substitution. Since

sec²x

has already been accounted for in the substitution, the integral reduces to:

∫ u⁹ du

. This is a straightforward power rule integration.

Step 5: Apply the power rule for integration. Recall that the integral of

uⁿ

(uⁿ⁺¹)/(n+1) + C

, where

C

is the constant of integration. After integrating, substitute back

u = tanx

to express the result in terms of

x

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Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, secant, and their inverses, are fundamental in calculus. They describe relationships between angles and sides of triangles and are periodic functions. Understanding their properties, such as identities and derivatives, is crucial for evaluating integrals involving these functions.

Integration Techniques

Integration techniques, including substitution and integration by parts, are essential for solving complex integrals. In the case of the integral ∫ tan⁹x sec⁴x dx, recognizing patterns and using appropriate substitutions can simplify the process. Mastery of these techniques allows for the effective evaluation of integrals that may not be straightforward.

Secant and Tangent Identities

The secant and tangent functions are related through the identity sec²x = 1 + tan²x. This relationship is particularly useful in integrals involving powers of tangent and secant, as it allows for the conversion between the two functions. Utilizing these identities can simplify the integration process and lead to a more manageable expression.

9–61. Trigonometric integrals Evaluate the following integrals.34. ∫ tan⁹x sec⁴x dx