Pattern 1: Cyclic Integrals integral e^xcos(x) dx: Let I = integral e^xcos(x) dx By parts: u = cos(x), dv = e^x dx I = e^xcos(x) + integral e^xsin(x) dx Apply by parts again to second integral: I = e^xcos(x) + e^xsin(x) - integral e^xcos(x) dx = e^xcos(x) + e^x*sin(x) - I 2I = e^x(cos(x) + sin(x)) I = e^x(cos(x) + sin(x))/2 + C

Pattern 2: Inverse Trig Functions integral arcsin(x) dx: u = arcsin(x), dv = dx = xarcsin(x) - integral x/sqrt(1-x^2) dx = xarcsin(x) + sqrt(1-x^2) + C

integral arctan(x) dx: u = arctan(x), dv = dx = xarctan(x) - integral x/(1+x^2) dx = xarctan(x) - (1/2)ln(1+x^2) + C

Pattern 3: Powers of x with transcendental functions integral x^3*e^x dx: Use tabular method. | Derivatives of x^3 | Integrals of e^x | Sign | | x^3 | e^x | + | | 3x^2 | e^x | - | | 6x | e^x | + | | 6 | e^x | - | | 0 | e^x | + | Result = e^x(x^3 - 3x^2 + 6x - 6) + C

Pattern 4: By parts producing original integral with different coefficient integral sec^3(x) dx: u = sec(x), dv = sec^2(x)dx I = sec(x)*tan(x) - integral sec(x)*tan^2(x) dx = sec(x)*tan(x) - integral sec(x)(sec^2(x)-1) dx = sec(x)*tan(x) - I + integral sec(x) dx 2I = sec(x)*tan(x) + ln|sec(x)+tan(x)| I = (1/2)[sec(x)*tan(x) + ln|sec(x)+tan(x)|] + C