Pattern 1: Cyclic Integrals integral $e^x$*cos(x) dx: Let I = integral $e^x$*cos(x) dx By parts: u = cos(x), dv = $e^x$ dx I = $e^x$*cos(x) + integral $e^x$*sin(x) dx Apply by parts again to second integral: I = $e^x$*cos(x) + $e^x$*sin(x) - integral $e^x$*cos(x) dx = $e^x$*cos(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 $\frac{x}{1+x^2}$ dx = xarctan(x) - $\frac{1}{2}$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 | | $x^3$ | $e^x$ | + | | 3$x^2$ | $e^x$ | - | | 6x | $e^x$ | + | | 6 | $e^x$ | - | | 0 | $e^x$ | + | Result = $e^x$($x^3$ - 3$x^2$ + 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 = $\frac{1}{2}$[sec(x)*tan(x) + ln|sec(x)+tan(x)|] + C