integration by substitution - Integration by Substitution SUGGESTED REFERENCE MATERIAL As you work through the problems listed below you should

integration by substitution - Integration by Substitution...

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Integration by Substitution SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 5.3 of the rec- ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS: Know how to simplify a “complicated integral” to a known form by making an appro- priate substitution of variables. PRACTICE PROBLEMS: For problems 1-21, evaluate the given indefinite integral and verify that your answer is correct by differentiation. 1. Z 3 x 2 ( x 3 + 3) 3 dx 2. Z 5 5 x + 3 dx 3. Z 2 x cos ( x 2 ) dx 4. Z 4 x ( x 2 + 6) 2 dx 5. Z sec (4 x ) tan (4 x ) dx 6. Z (3 x - 5) 9 dx 7. Z e - 2 x dx 8. Z sin x cos x 1 + sin 2 x dx 9. Z csc x 2 cot x 2 dx 10. Z - 3 x 3 1 - x 4 dx 11. Z e x x - 1 4 cos 4 x dx 1
2/3dx19.Zcsc2(3x) tan2(3x) +x2ex3dx20.Z1xlnxdx21.Zcos-1x1-x2dx Z 1 22. Use an appropriate trigonometric identity followed by a reasonable substitution toevaluateZtanx dx2+ 77x+ 4921 23. It can be shown that32x(3x+ 1)(4x+ 5)2=3x+ 1-(4x+ 5) 2. Use this fact to evaluateZ32x2+ 77x+ 49(3x+ 1)(4x+ 5)2dx.24. Using the substitutionx= sinθfor-π2θπ2, evaluateZ1-x2dx. Expressyour answer completely in terms of the variablex. HINT - The following trigonometric identities will be helpful:sin2θ+ cos2θ= 1,cos2θ=12(1 + cos (2θ), and sin (2θ) = 2 sinθcosθ 2

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