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Illustration: a) b) c)
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3. Illustration: a) b)
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4. Illustrations: a) may be deleted may be deleted b) may be deleted
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BASIC THEOREMS OF INTEGRATION Let: u, v, w - are integrable functions of x 1. 2. 3. FUNDAMENTAL METHODS OF INTEGRATION I. THE POWER INTEGRAL: II. THE GENERAL POWER INTEGRAL:
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1. 2. 3. 4. 5. SUMMARY:
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Exercises: Evaluate the following integrals. 1. 8. 2. 9. 3. 10. 4. 11. 5. 12. 6. 13. 7. (2x +3)dx 14. 1. 2. 3. 4. 5.
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Simple Substitution (General Power Formula) A technique called substitution , that can often be used to transform complicated integration problems into simpler ones. Quite often, the process of integration can be simplified by use of a substitution or change of variable. The purpose of substituting a new variable is to bring the problem to a form for which the standard formula, = + C n can be applied.
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Application: Integrate the following using u-substitution. 1. Solution. Let Substitute to the given, that lead to (Note: is known as integrating factor or correction factor) Apply general power formula Substitute back the value of u, the final answer is,
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2. 3. Let u = Let u du = du = = = (Recall the law of exponents) =
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EXAMPLES: Find the indefinite integral. 1. 2. 3. 4. 5.
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