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AP Calculus AB: 9.3.3 More Integrating Tirgonometric Functions by Substitution

Mathematics12 CardsCreated 3 months ago

This section explores advanced uses of integration by substitution with trigonometric functions, even when they are not composite. It emphasizes selecting appropriate substitutions, applying trigonometric identities, and simplifying expressions to integrable forms. The examples demonstrate how to handle products of trig functions and logarithmic results from integrals like ∫ cot 𝑥 𝑑𝑥.

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More Integrating Trigonometric Functions by Substitution

  • You can apply integration by substitution to integrands involving trigonometric functions that are not composite functions.

  • When working with integrands that include trigonometric expressions, it is sometimes necessary to rewrite those expressions using trig identities.

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Key Terms

Term
Definition

More Integrating Trigonometric Functions by Substitution

  • You can apply integration by substitution to integrands involving trigonometric functions that are not composite functions.

  • ...

note

  • Instead of a composite function, this integral involves the product of two trigonometric functions.

  • You could let u be sinx,...

Evaluate the integral.

∫secxtanx√1+secxdx

2/3(1+secx)^(3/2)+C

Integrate.∫cotxdx

ln | sin x | + C

Integrate.∫csc^2t / tan^2t dt

−cot^3t/3+C

Evaluate.∫8sin^33xcos3xdx

2/3sin^4 3x+C

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AP Calculus AB: 9.3.3 More Integrating Tirgonometric Functions by Substitution
TermDefinition

More Integrating Trigonometric Functions by Substitution

  • You can apply integration by substitution to integrands involving trigonometric functions that are not composite functions.

  • When working with integrands that include trigonometric expressions, it is sometimes necessary to rewrite those expressions using trig identities.

note

  • Instead of a composite function, this integral involves the product of two trigonometric functions.

  • You could let u be sinx, in which case du would be cosx, or you could let u be cosx, making du be –sinx. You might want to choose u = sinx to avoid the negative sign.

  • Once you have determined the expression for u, the integrand should be simple to evaluate. Remember to replace u with its expression in terms of x.

  • You can check your work by integrating with the help of the chain rule.

  • You may often find it useful to express trigonometric
    integrands in terms of the sine and cosine functions.

  • Notice that you must choose u = cosx, since it is in the denominator. That way the du-expression can replace the numerator and dx.

  • Factor out the –1 from the integrand.

  • The integral of du/u is ln|u| + C.

  • Make sure to express your result in terms of x.

  • Check that your answer is correct by integrating.

Evaluate the integral.

∫secxtanx√1+secxdx

2/3(1+secx)^(3/2)+C

Integrate.∫cotxdx

ln | sin x | + C

Integrate.∫csc^2t / tan^2t dt

−cot^3t/3+C

Evaluate.∫8sin^33xcos3xdx

2/3sin^4 3x+C

Integrate.∫cos^4(x^2)sin(x^2)2xdx

−cos^5x^2/5+C

Integrate.∫tanxdx

ln | sec x | + C

Evaluate.∫tan^2θ/cos^2θdθ

tan^3θ/3+C

Integrate.∫sec(sinx)tan(sinx)cosxdx

sec (sin x) + C

Evaluate.∫sinxcosxdx

sin^2x/2+C

Evaluate the integral.∫cosxcos(sinx) dx

sin (sin x) + C