Chapter 7 – Techniques of Integration 7.3 Trigonometric Substitution 1Erickson

When do we use it? 7.3 Trigonometric Substitution2  Trigonometric substitution is used when you have problems involving square roots with two terms under the radical.  You will make one of the substitutions below depending on what is inside your radical. Erickson ExpressionSubstitution Identity

General Process 7.3 Trigonometric Substitution3 1. Decide which formulas you will work with. 2. Draw your triangle. 3. Solve the triangle. 4. Make your trig substitutions. 5. Integrate. 6. Convert back to the original variables. Erickson

Example 1 1. Choose your formula. This matches the first set of formulas, so x = 3sin  2.Draw your triangle sin  = x/3 x 3 3. Solve the triangle 4. Make your trig substitutions Trigonometric SubstitutionErickson

5. Integrate 6. Convert Back x 3 Example 1 continued 57.3 Trigonometric SubstitutionErickson

Example 2 1. Choose your formula. This matches the third set of formulas, so x = 2sec  2.Draw your triangle sec  = x/2 2 x 3. Solve the triangle 4. Make your trig substitutions Trigonometric SubstitutionErickson

5. Integrate 6. Convert Back Example 2 continued 2 x 77.3 Trigonometric SubstitutionErickson

Example 3 1. Choose your formula. This matches the first set of formulas, so 2x = 4tan  2.Draw your triangle tan  = (2x)/4 2x2x 4 3. Solve the triangle 4. Make your trig substitutions Trigonometric SubstitutionErickson

5. Integrate 6. Convert Back Example 3 continued 97.3 Trigonometric SubstitutionErickson

Examples 7.3 Trigonometric Substitution10  Evaluate the integral. Erickson

Example Erickson7.3 Trigonometric Substitution11  Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves