Integrals Involving Exponential and Logarithmic Functions

Integral of the Exponential function

Since

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}e^x=e^x}

we see that $e^{x}$ is its own antiderivative. This allows us to find the integral of an exponential function:

$\int e^{x}dx=e^{x}+C$

Integral of the Inverse function

To integrate $f(x)={\frac {1}{x}}$ , we should first remember

${\frac {d}{dx}}\ln(x)={\frac {1}{x}}$

Therefore, since ${\frac {1}{x}}$ is the derivative of $\ln(x)$ we can conclude that

Note that the polynomial integration rule does not apply when the exponent is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} . This technique of integration must be used instead. Since the argument of the natural logarithm function must be positive (on the real line), the absolute value signs are added around its argument to ensure that the argument is positive.

Resources

Integrating Exponential Functions By Substitution - Antiderivatives - Calculus by The Organic Chemistry Tutor