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INTEGRATION

INTEGRATION. TECHNIQUES OF INTEGRATION 1.2. TECHNIQUES OF INTEGRATION. Integration By Substitution. Integration By Partial Fraction. Integration By Substitution. Step 1 : Substitute u=g(x) , du= g’(x) dx to obtain the integral

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INTEGRATION

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  1. INTEGRATION TECHNIQUES OF INTEGRATION 1.2

  2. TECHNIQUES OF INTEGRATION • Integration By Substitution • Integration By Partial Fraction

  3. Integration By Substitution Step 1 : Substitute u=g(x), du= g’(x) dx to obtain the integral Step 2 : Integrate with respect to u Step 3 : Replace uby g(x) in the result

  4. *The integranddoes not have the same variable with the variable of integration.

  5. Example Find the integration of Step 1 : Substitute u = g(x), du = g’(x) dx to obtain the integral

  6. Step 2 : Integrate with respect to u

  7. Step 3 : Replace uby g(x) in the result

  8. Example 1 Example 2 Example 4 Example 3

  9. Example 5 Example 6 Example 7 By using the suitable substitution, find

  10. Example 8 Use the given substitution, find Example9 • Integrate the following with respect to x:

  11. EXERCISE 1 Find the following integrals. Answer

  12. Integration By Partial Fractions To integrate a rational function, it can be express in terms of its partial fraction. P(x) Example q(x)

  13. The Rules • The numerator of a given function must be of lower degree than that denominator (mean that the function is a proper fraction).

  14. Tips! Linear Factor Repeated Linear Factor Quadratic Factor Repeated Quadratic Factor

  15. Example 1 By using partial fractions, find

  16. Example 2

  17. Exercise 1 • Express in the form of partial fractions. Hence, find Answer: 4 ln (x -1) – 2ln(x 2 + 2) + C

  18. Exercise 2 Use partial fractions to find Answer:

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