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:

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download