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

Integrating exponential & logarithmic functions

Exponential functions have the general form $y=a^x$.  Special case:  $y={\mathrm{e}}^x$.

Logarithmic functions have the general form $y={\log }_{a}x$.  Special case:  $y={\log }_{\mathrm{e}}x=\ln x$.

What are the antiderivatives of exponential and logarithmic functions?

• Those involving the special cases have been met before

  • ${\int }_{}^{}{\mathrm{e}}^{x}dx={\mathrm{e}}^x+c$

  • ${\int }_{}^{}\frac{1}{x}dx=\ln \left|x\right|+c$

  • These are given in the formula booklet

• Also

  • ${\int }_{}^{}{a}^{x}dx=\frac{1}{\ln a}{a}^x+c$

  • This is also given in the formula booklet

• By reverse chain rule

  • ${\int }_{}^{}\frac{1}{x\ln a}dx={\log }_a\left|x\right|+c$

  • This is not in the formula booklet

    • but the derivative of ${\log }_{a}x$ is given

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

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

  • $\int \frac{f^{\text{'}}\left(x\right)}{f\left(x\right)}dx=\ln \left|f\left(x\right)\right|+c$

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

• For the special cases involving $e$ and $\ln$

  • ${\int }_{}^{}{\mathrm{e}}^{ax+b}dx=\frac{1}{a}{\mathrm{e}}^{ax+b}+\mathrm{c}$

  • $\int \frac{1}{ax+b}dx=\frac{1}{a}\ln \left|ax+b\right|+c$

• For the general cases

  • $\int a^{px+q}\mathrm{d}x=\frac{1}{p\ln a}{a}^{px+q}+c$

  • $\int \frac{1}{\left(px+q\right)\ln a}\mathrm{d}x=\frac{1}{p}{\log }_a|px+q|+c$

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