Some important definitions:
|Some key results (to learn): | |You need to remember these trigonometric formulae- they are needed in |
|[pic][pic] | |some integration questions. |
|[pic] |[pic] | |
|[pic] | |[pic] [pic] |
| | |[pic] [pic] |
| | |[pic] |
|Integration by substitution |Methods of integration |Integration involving trigonometrical identities |
| | | |
| |The methods of integrating an expression are: |Example: Using the identity given above [pic] |
| |1) directly writing it down (either by memory or by looking in the | |
| |formula book); |Example 2: Find [pic] |
| |2) writing the expression in partial fractions; |Solution: We use the fact that [pic] |
| |3) using the method of integration by parts; |So, [pic] |
| |4) using a substitution; |= [pic] |
| |5) using a trigonometric identity (such as [pic] or [pic] ) |=[pic] |
| | | |
| |See my separate sheet on how to identify which method of integration is |We now need to work out [pic] |
| |the appropriate one to use in any given situation. |We use the substitution |
| | |[pic] |
| | |So, [pic] |
| | |=[pic] |
| | | |
| | |Therefore, [pic] |
| | |= [pic] |
|Example: [pic]. | | | |
|Make the substitution [pic] | | | |
|We get: [pic] |[pic] | | |
|This gives: [pic] |[pic] | | |
|Example 2: Use the substitution [pic] to find | | | |
|[pic]. | | | |
| | | | |
|Solution: [pic] | | | |
|Since [pic], |[pic] | | |
|[pic] |[pic] | | |
|So we get [pic] |[pic] | | |
| | |Definite integrals using a substitution | |
| | |Find [pic]. | |
| | |Use the substitution [pic]. |[pic] | |
| | |[pic] | | |
| | |= [pic] |x = 0 → u = 0 | |
| | | |x = 2 → u = 4 | |
|Take the multipliers outside the integral: | | | |
|[pic] | | | |
|This gives: [pic] | | | |
|But [pic], so [pic]. | | | |
|This expands to give: [pic] | | | |
-----------------------
This column gives the calculations for changing the dx to du:
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