Exponential Integrals:

integral e^(ax)sin(bx) dx = e^(ax)(asin(bx) - bcos(bx))/(a^{2+b}^2) + C integral e^(ax)cos(bx) dx = e^(ax)(acos(bx) + bsin(bx))/(a^{2+b}^2) + C (Derived by applying by parts twice and solving for I)

integral $x^n$ * $e^x$ dx: use by parts n times (tabular method) integral $e^x$ * $x^n$ = $e^x$ * ($sum_{k=0}^{n}$ (-1)^k * n!/(n-k)! * x^(n-k)) + C

Logarithmic Integrals:

integral ln(x) dx = xln(x) - x + C integral (ln(x))^2 dx = x(ln(x))^2 - 2xln(x) + 2x + C integral $x^n$ * ln(x) dx = x^(n+1)[ln$\frac{x}{(n+1)}$ - $\frac{1}{n+1}$^2] + C (for n != -1) integral ln$\frac{x}{x}$ dx = (ln(x))^2/2 + C (substitution u = ln(x))

Special Exponential Forms:

integral e^(sqrt(x)) dx: Let t = sqrt(x), x = $t^2$, dx = 2t dt = 2integral t$e^t$ dt = 2$e^t$(t-1) + C = 2e^(sqrt(x))(sqrt(x)-1) + C

integral $e^x$ * $\frac{x-1}{x}$^2 dx: Write $\frac{x-1}{x}$^2 = 1/x - 1/$x^2$. Note d/dx$\frac{1}{x}$ = -1/$x^2$. So this is $e^x$[f(x) + f'(x)] with f = 1/x... wait: f(x) + f'(x) = 1/x + (-1/$x^2$) = 1/x - 1/$x^2$ = $\frac{x-1}{x}$^2. Yes! integral = $e^x$/x + C