Exponential and Logarithmic Integrals

The integration rules for exponential functions are the inverse of their differentiation rules.
Simple Exponential Rule: $\int e^x \,dx = e^x + C$
General Exponential Rule: Let $u$ be a differentiable function of $x$ . $\int e^u \frac{du}{dx} \,dx = \int e^u \,du = e^u + C$
Integration Examples:
- $\int 2e^x \,dx = 2e^x + C$
- $\int 2e^{2x} \,dx$ : Let $u = 2x$ , then $\frac{du}{dx} = 2$ . Result: $e^{2x} + C$
- $\int (e^x + x) \,dx = e^x + \frac{x^2}{2} + C$
- $\int 5x e^{-x^2} \,dx$ : Let $u = -x^2$ , then $\frac{du}{dx} = -2x$ . Factoring out constants yields $-\frac{5}{2} \int e^u \,du = -\frac{5}{2} e^{-x^2} + C$

The Log Rule accounts for cases where the Power Rule for integration fails ( $n = -1$ ).
General Log Rule: $\int \frac{1}{u} \frac{du}{dx} \,dx = \int \frac{du}{u} = \ln|u| + C$
Integration Example:
- $\int \frac{6x}{x^2 + 1} \,dx$ : Factoring out a $3$ to create the numerator $2x$ (the derivative of $x^2 + 1$ ) results in $3 \ln(x^2 + 1) + C$ .

Some integrands must be algebraically manipulated before applying the Exponential or Log Rules:
Splitting Fractions: $\int \frac{3x^2 + 2x - 1}{x^2} \,dx = \int (3 + \frac{2}{x} - \frac{1}{x^2}) \,dx = 3x + 2 \ln |x| + \frac{1}{x} + C$
Multiplying by a Conjugate or Factor: To find $\int \frac{1}{1 + e^{-x}} \,dx$ , multiply the numerator and denominator by $e^x$ to get $\int \frac{e^x}{e^x + 1} \,dx = \ln(e^x + 1) + C$ .
Polynomial Division: For rational functions where the degree of the numerator is greater than or equal to the denominator: $\int \frac{x^2 + x + 1}{x - 1} \,dx = \int (x + 2 + \frac{3}{x - 1}) \,dx = \frac{x^2}{2} + 2x + 3 \ln |x - 1| + C$