Trick 1: King's Rule Substitution for Symmetric Functions $integral_0^a$ f(x) dx = $integral_0^a$ f(a-x) dx (definite integral property, but useful to recognize pattern). For indefinite: if f(x) + f(a-x) = constant, the integral simplifies dramatically.

Trick 2: Multiplying and Dividing integral sec(x) dx: multiply by (sec(x)+tan(x))/(sec(x)+tan(x)) = integral ($sec^2$(x)+sec(x)tan(x))/(sec(x)+tan(x)) dx Substitute u = sec(x)+tan(x), du = (sec(x)tan(x)+$sec^2$(x))dx = ln|sec(x)+tan(x)| + C

Trick 3: Adding and Subtracting in Numerator integral $\frac{x}{x+1}$ dx = integral ((x+1)-1)/(x+1) dx = integral (1 - $\frac{1}{x+1}$) dx = x - ln|x+1| + C

Trick 4: Half-angle for $\frac{1}{a+b*cos(x}$) type integral $\frac{dx}{5+4cos(x}$): Let t = tan$\frac{x}{2}$. cos(x) = $\frac{1-t^2}{(1+t^2)}$, dx = 2$\frac{dt}{1+t^2}$ = integral (2$\frac{dt}{1+t^2}$)/$\frac{5 + 4(1-t^2}{(1+t^2)}$) = integral 2$\frac{dt}{5+5t^2+4-4t^2}$ = integral 2$\frac{dt}{9+t^2}$ = $\frac{2}{3}$arctan$\frac{t}{3}$ + C = $\frac{2}{3}$arctan$\frac{tan(x/2}{3}$) + C

Trick 5: Substitution for integral $\frac{dx}{x(x^n+1}$) Multiply numerator and denominator by x^(n-1): = integral x^(n-1)$\frac{dx}{x^n(x^n+1}$) Let t = $x^n$, dt = nx^(n-1)dx = $\frac{1}{n}$*integral $\frac{dt}{t(t+1}$) = $\frac{1}{n}$*integral (1/t - $\frac{1}{t+1}$) dt = $\frac{1}{n}$*ln|$\frac{t}{t+1}$| + C = $\frac{1}{n}$*ln|x^$\frac{n}{x^n+1}$| + C

Trick 6: Rationalizing Substitution integral $\frac{dx}{1+sqrt(x}$): Let t = sqrt(x), x = $t^2$, dx = 2t dt = integral 2t $\frac{dt}{1+t}$ = 2*integral (1 - $\frac{1}{1+t}$) dt = 2t - 2ln|1+t| + C = 2sqrt(x) - 2ln(1+sqrt(x)) + C