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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes5. CalculusFurther IntegrationIntegrating with Exponential & Logarithmic Functions

Integrating with Exponential & Logarithmic Functions (DP IB Analysis & Approaches (AA)): Revision Note

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Integrating Exponential & Logarithmic Functions

Exponential functions have the general form y equals a to the power of x.  Special case:  y equals straight e to the power of x.

Logarithmic functions have the general form y equals log subscript a x.  Special case:  y equals log subscript straight e x equals ln space x.

What are the antiderivatives of exponential and logarithmic functions?

  • Those involving the special cases have been met before

    • integral subscript blank superscript blank straight e to the power of x space d x equals straight e to the power of x plus c

    • integral subscript blank superscript blank 1 over x space d x equals ln space vertical line x vertical line plus c

    • These are given in the formula booklet

  • Also

    • integral subscript blank superscript blank a to the power of x space d x equals fraction numerator 1 over denominator ln space a end fraction a to the power of x plus c

    • This is also given in the formula booklet

  • By reverse chain rule

    • integral subscript blank superscript blank fraction numerator 1 over denominator x ln space a end fraction space d x equals log subscript a vertical line x vertical line plus c

    • This is not in the formula booklet

      • but the derivative of log subscript a x is given

  • There is also the reverse chain rule to look out for

    • this occurs when the numerator is (almost) the derivative of the denominator

    • integral subscript blank superscript blank fraction numerator f apostrophe left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis end fraction space d x equals ln space open vertical bar space f left parenthesis x right parenthesis space close vertical bar plus c

How do I integrate exponentials and logarithms with a linear function of x involved?

  • For the special cases involving e and ln

    • integral subscript blank superscript blank straight e to the power of a x plus b end exponent space d x equals 1 over a straight e to the power of a x plus b end exponent plus straight c

    • integral subscript blank superscript blank fraction numerator 1 over denominator a x plus b end fraction space d x equals 1 over a ln space open vertical bar space a x plus b close vertical bar plus c

  • For the general cases

    • integral a to the power of space p x plus q end exponent space straight d x equals fraction numerator 1 over denominator p space ln space a end fraction a to the power of space p x plus q end exponent plus c

    • integral fraction numerator 1 over denominator open parentheses p x plus q close parentheses ln space a end fraction space straight d x equals 1 over p log subscript a vertical line p x plus q vertical line plus c

  • These four results are not in the formula booklet but all can be derived using ‘adjust and compensate’ from reverse chain rule

Examiner Tips and Tricks

  • Remember to always use the modulus signs for logarithmic terms in the antiderivative

    • Once it is deduced that  g left parenthesis x right parenthesis in  ln space vertical line g left parenthesis x right parenthesis vertical line, say, is guaranteed to be positive, the modulus signs can be replaced with brackets

Worked Example

a)       Show that integral subscript 1 superscript 2 4 to the power of x space d x equals fraction numerator 6 over denominator ln space 2 end fraction.

5-9-1-ib-hl-aa-only-we3a-soltn

b)       Find integral fraction numerator 1 over denominator open parentheses 2 x minus 1 close parentheses space ln space 3 end fraction space straight d x.

zgf4~aK9_5-9-1-ib-hl-aa-only-we3b-soltn

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    Paul

    Author: Paul

    Expertise: Maths Content Creator

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.