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 since the natural logarithm function is only defined for positive numbers.

Problems

Integrals involving Exponents of Eulers constant

Problem 21

Evaluate the following integral \int e^x \, dx

Problem 22

Evaluate the following integral \int 5e^{5x} \, dx

Problem 23

Evaluate the following integral \int e^{2x + 2} \, dx

Problem 24

Evaluate the following integral \int \frac{1}{e^x} \, dx

Problem 25

Evaluate the following integral \int \frac{e^{2x} + 4}{e^{2x}} \, dx

Integrals involving Exponents of Arbitrary constants

Problem 26

Evaluate the following integral \int a^x \, dx

Problem 27

Evaluate the following integral \int a^{bx} \, dx

Problem 28

Evaluate the following integral \int a^{2x - 2} \, dx

Integrals involving 1/x

Problem 29

Evaluate the following integral \int \frac{1}{x} \, dx

Problem 30

Evaluate the following integral \int \frac{1}{x + 5} \, dx

Problem 31

Evaluate the following integral \int \frac{x + 5}{x^2 + 10x + 25} \, dx