Logarithms and Exponents

Next we move onto logarithms and exponents.

Formulas

It is well known that e^x is it’s own derivative, giving us the formula

\int e^x \, dx = e^x + C.

Using chain rule, we get some slightly modified formulas.

Theorem

The following integrals hold \begin{align*} &\int e^x \, dx&&= e^x + C \\ &\int ae^{ax} \,dx &&= e^{ax} + C\\ &\int e^{ax + b} \,dx &&= \frac{1}{a}e^{ax + b} + C \end{align*}

But these are just raising the constant e to a power, what if we have some arbitrary constant a?

Theorem

The following integrals hold \begin{align*} &\int a^x \, dx&&= \frac{a^x}{\ln a} + C \\ &\int ba^{bx} \,dx &&= \frac{a^{bx}}{\ln a} + C\\ &\int a^{bx + c} \,dx &&= \frac{a^{bx+c}}{b\ln a} + C \end{align*}

Next we look at logarithms. This is a real simple one

Theorem

This integral holds \begin{align*} &\int \frac{1}{x} \, dx&&= \ln |x| + C \\ &\int \frac{1}{x + a} \,dx &&= \ln |x| + C\\ \end{align*}

These absolute value bars are necessary for some reason I don’t fully get it.