IB Diploma programme – HL Mathematics - Summary of ...



IB Diploma Programme – HL Mathematics - Summary of Integration Techniques

|Polynomials and other power functions |Further examples and notes |

|[pic] Where [pic]and [pic]are real numbers (constants) and [pic] |[pic] |

|Increase the old power by 1 to get the new power and divide by the new power. |[pic] |

|Basic trigonometric functions | |

|These are standard integrals – see formula sheet. |Divide by [pic] to undo the effect of the chain rule. |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|More trig integrals – two special cases | |

|[pic][pic] |Rearrange the following trig identities for [pic] |

| |[pic] |

| |[pic] |

| |to obtain simpler expressions for [pic]and [pic] |

|Integrating exponential functions | |

|If the base is [pic]then |If the base is any other positive number, [pic], then |

|[pic] |[pic] |

|[pic] |[pic] |

|Integrating [pic]and [pic] | |

|[pic] or [pic] |When solving differential equations, it is often helpful to replace the constant of |

|[pic] |integration [pic]with another constant in the form [pic], therefore we can use log |

| |rules to write [pic] |

|Integrating products using the chain rule in reverse | |

|Look out for this pattern. The integral of a (main or outer) function of another |Ignore[pic]and integrate[pic] to get [pic] |

|(inner) function with the derivative of the inner function present as a multiple.| |

| | |

|The answer is the integral of the main (outer function) |Here are a few more examples: |

|[pic] | |

|[pic][pic] | |

|[pic][pic] |[pic] |

|You can memorise this pattern (and use it correctly) Or use the substitution |[pic] |

|[pic]to obtain the required result. |[pic] |

| |Always check your answer using differentiation. |

|Integrating products using algebraic substitution |Examples |

|If the pattern above is not present then use a suitable algebraic substitution. |1.[pic] using [pic] we can transform the integral into [pic]which is much easier to |

| |integrate than the original question. Remember to write the final answer in terms of |

|Here is a list of Some helpful substitutions |[pic] |

| |i.e [pic] |

|If the integrand contains Use |2.[pic]use [pic] to get [pic] |

|[pic] |and finally we get [pic] |

| |3.[pic] using [pic]we get [pic] |

| |And finally we get [pic] |

| | |

| | |

| | |

|Integrating quotients | |

|[pic] |[pic] |

|Integrals of the form[pic]should be simplified using long division. Otherwise use| |

|substitution. |[pic] |

|Integration by parts | |

|Some integrals can not be done by the above strategies |The method of integration by parts can only be attempted if the function chosen as |

|Examples are [pic] and [pic] |[pic]is easy to integrate. |

|The above integrals can be done using the formula |When solving [pic], we choose |

|[pic] |[pic] and |

|Make sure that the integral [pic] is easier to tackle than the original integral.|[pic] then |

|When solving [pic], we choose |[pic] |

|[pic] and |[pic] |

|[pic] then | |

|[pic] | |

|[pic] | |

| | |

|Check the answer by differentiation | |

[pic]

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