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Trigonometric Substitution Notes

Trigonometric Substitution Notes - Trigonometric...

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Trigonometric Substitution (Section 7.3) Brief Trigonometry Review: Reference Triangles: sin θ = sec θ = cos θ = csc θ = tan θ = cot θ = Example 0.1. If cos θ = x 4 , find the values of the remaining 5 trig functions. Trigonometric Substitution Used for integrals of the form: R dx ( a 2 + x 2 ) m R dx ( a 2 - x 2 ) m R dx ( x 2 - a 2 ) m R a 2 + x 2 dx R a 2 - x 2 dx R x 2 - a 2 dx
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Use the following substitutions: Expression Substitution Identity a 2 - x 2 x = a sin θ 1 - sin 2 θ = cos 2 θ a 2 + x 2 x = a tan θ 1 + tan 2 θ = sec 2 θ x 2 - a 2 x = a sec θ sec 2 θ - 1 = tan 2 θ Example 0.2. Using trig substitution for indefinite integrals: Evaluate Z dx 9 + x 2 1. Determine which of the above substitutions to use. 2. Find dx . 3. Integrate in terms of θ using whatever method is appropriate. 4. Use a reference triangle to replace trig functions with expressions in terms of x .
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Example 0.3. Evaluate Z x 2 4 - x 2 dx Example 0.4. Trigonometric Substitution for Definite Integrals: Evaluate Z 4 2 x 2 - 4 x dx 1. Integrate as an indefinite integral: Z
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