# n8b - Section 8.3 Trigonometric Substitution courtesy...

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Unformatted text preview: Section 8.3 Trigonometric Substitution courtesy: Chia-Rong Chen 1. When the integrand involves a 2- x 2 , a 2 + x 2 , or x 2- a 2 , and the integral cannot be evaluated by substitution method, we make use of the identities: 1- sin 2 θ = cos 2 θ, 1 + tan 2 θ = sec 2 θ, sec 2 θ- 1 = tan 2 θ. 2. trigonometric substitutions: expression substitution range new expression final substitution a 2- x 2 x = a sin θ- π 2 ≤ θ ≤ π 2 a 2 cos 2 θ θ = sin- 1 x a a 2 + x 2 x = a tan θ- π 2 < θ < π 2 a 2 sec 2 θ θ = tan- 1 x a x 2- a 2 x = a sec θ ≤ θ < π 2 , a 2 tan 2 θ θ = sec- 1 x a or π ≤ θ < 3 π 2 eg.(1) Evaluate integraldisplay dx x 2 √ x 2- 16 dx . 1 eg.(2) integraldisplay dx a 2 + x 2 eg.(3) integraldisplay x 2 √ 9- x 2 dx 2 eg.(4) integraldisplay 1 x 3 √ x 2 + 1 dx eg.(5) integraldisplay dx √ x 2 + 4 x + 8 3 . eg.(6) integraldisplay √ e 2 t- 9 dt 4 eg.(7) integraldisplay √ a 2- x 2 dx 5 Section 8.4 Integration of Rational Functions by Partial Fractions...
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n8b - Section 8.3 Trigonometric Substitution courtesy...

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