Formulas and Tables

Calculus

Integrals of Exponential and Logarithmic Functions

Functions: \({e^x},\) \({a^x},\) \(\ln x,\) \(\sin x,\) \(\cos x\)
Argument (independent variable): \(x\)

Natural number: \(n\)
Real numbers: \(C\), \(a\), \(b\)

  1. Integral of the exponential function
    \({\large\int\normalsize} {{e^x}dx} = {e^x} + C\)
  2. Integral of the exponential function with base a
    \({\large\int\normalsize} {{a^x}dx} = {\large\frac{{{a^x}}}{{\ln a}}\normalsize} + C,\) \(a \gt 0.\)
  3. \({\large\int\normalsize} {{e^{ax}}dx} = {\large\frac{{{e^{ax}}}}{{a}}\normalsize} + C,\) \(a \ne 0.\)
  4. \({\large\int\normalsize} {x{e^{ax}}dx} =\) \({\large\frac{{{e^{ax}}}}{{{a^2}}}\normalsize}\left( {ax – 1} \right) + C,\) \(a \ne 0.\)
  5. Integral of the natural logarithm
    \({\large\int\normalsize} {\ln x\,dx} =\) \( x\ln x – x + C\)
  6. \({\large\int\normalsize} {\large\frac{{dx}}{{x\ln x}}\normalsize} =\) \( \ln \left| {\ln x} \right| + C\)
  7. \({\large\int\normalsize} {{x^n}\ln x\,dx} =\) \( {x^{n + 1}}\left[ {{\large\frac{{\ln x}}{{n + 1}}\normalsize} – {\large\frac{1}{{{{\left( {n + 1} \right)}^2}}}}\normalsize} \right] \) \(+\; C\)
  8. \({\large\int\normalsize} {{e^{ax}}\sin {bx}\,dx} =\) \( {\large\frac{{a\sin {bx} – b\cos {bx}}}{{{a^2} + {b^2}}}\normalsize} {e^{ax}} + C\)
  9. \({\large\int\normalsize} {{e^{ax}}\cos {bx}\,dx} =\) \({ \large\frac{{a\cos {bx} + b\sin {bx}}}{{{a^2} + {b^2}}}\normalsize} {e^{ax}} + C\)