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A2 Differentiation and Integration Methods. Differentiation Function yx n y ax n y ax n bx m y (ax n b)m Differential dy dx dy dx dy dx dy dx nx n 1 anx n 1 anx n 1 bmx m 1 manx n 1 (ax n b) m 1 y ( f ( x))n dy nf '( x)( f ( x)) n 1 dx y ln x dy 1 dx x dy f '( x) dx f ( x) dy ex dx y ln f ( x) y ex y e f ( x) y ax y sin x y cos x dy dx dy dx dy dx dy dx f '( x)e f ( x ) Example dy y x5 5x4 dx dy y 3x 4 12 x3 dx dy y 2 x 4 5 x3 8 x 3 15 x 2 dx dy y (2 x3 4)5 30 x 2 (2 x3 4) 4 dx y (2 x5 3x7 )3 3(10 x 4 21x6 )(2 x5 3x7 )2 y ln 5 x 2 y e2 x 3 dy 10 x 2 dx 5 x 2 x 3 dy 6 x 2e2 x dx a x ln a cos x sin x 1 Notes If you differentiate a constant you get zero You can differentiate polynomials term by term. Multiply by the power of the bracket. Multiply by the differential of the function in the bracket. Drop the power of the bracket by 1. This is the general formula that works for polynomials and later functions. (E.g. see sinnf(x)) Function differentiated divided by function. A special function whose gradient at any point is equal to the value of the function at that point. e to the function of x multiplied by the differential of the function. When a=e, lne=1 so result reduces to that above for ex. A2 Differentiation and Integration Methods. y tan x y sin nx y cos nx y tan nx y sin f ( x) y cos f ( x) y tan f ( x) y sec x y cot x y cos ecx y sin n f ( x) y cos n f ( x) y tan n f ( x) y sin 1 x dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx sec 2 x n cos nx n sin nx n sec 2 nx f '( x) cos f ( x) f '( x) sin f ( x) f '( x) sec 2 f ( x) dy 3cos 3 x dx dy y cos 5 x 5sin 5 x dx dy y tan 2 x 2sec 2 2 x dx dy y sin 2 x 7 14 x 6 cos 2 x 7 dx dy y cos 3x 6 18 x5 sin 3 x 6 dx dy y tan 3x 4 12 x3 sec 2 3x 4 dx y sin 3x Differential of function multiplied by cos of function. Differential of function multiplied by differential of sin of function. Differential of function multiplied by differential of tan of function. sec x tan x cos ec 2 x cos ecx cot x nf '( x) cos f ( x) sin n 1 f ( x) y sin 5 3x 2 dy 30 x cos 3x 2 sin 3x 2 dx dy nf '( x) sin f ( x) cos n 1 f ( x) dx dy nf '( x) sec 2 f ( x) tan n 1 f ( x) dx dy 1 dx 1 x2 2 Multiply by the power. Multiply by the differential of the function Multiply by the differential if sin. Drop the power by one. A2 Differentiation and Integration Methods. dy 1 dx 1 x2 dy 1 dx 1 x 2 dy dv du u v dx dx dx du dv v u dy dx 2 dx dx v y cos1 x y tan 1 x y uv y u v dy x cos x sin x dx x dy sin x x cos x y sin x dx sin 2 x y x sin x Product rule Quotient rule. Integration Function Integral x dx x c n 1 ax dx ax n 1 c n 1 ax n 1 bx m1 c n 1 m 1 (ax n b)m c n ax n bx m dx manx n 1 (ax n b)m1dx nf '( x)( f ( x)) 1 x dx Example n 1 n n 1 dx ( f ( x))n c x dx 3 4 x c 4 2 x5 5 x 4 2 x 5x dx 5 4 c 2 3 4 3 5 30 x (2 x 4) dx (2 x 4) c 4 3(10 x 3 4 21x 6 )(2 x5 3x 7 ) 2 dx (2 x5 3x 7 )3 c ln x c 3 Notes Raise the power by 1 and divide by the new power. Does not work when n=-1, this give the special case lnx. You can integrate polynomials term by term. Realise that the differential of the bracket is outside the bracket. This is the general formula. Realise that the differential of the bracket is outside the bracket. x0 Consider x 1dx c using the 0 basic rule for integration. This would imply that the area is always infinite beneath a 1/x graph, which is clearly ridiculous. Hence the A2 Differentiation and Integration Methods. ln f ( x) c f '( x) dx f ( x) e dx ex c e eax a ax c ln a x ln x x c x ax dx a dx x ln xdx sin xdx cos xdx tan xdx sin nxdx 10 x 2 5x 2 dx ln 5 x 3 x e dx e3 x c 3 special case. Spot, function differentiated divided by function. Since the differential of ex is ex, then the integral of ex is ex (+c). e to the ax divided by a.. If a=e this reduces to the ex result above. cos x c sin x c ln sec x c cos nxdx sec xdx cot xdx cos ecxdx ln can only take positive values. cos nx c n sin 3xdx cos 3 x c 3 sin nx c n ln s ec x tan x c cos 5xdx sin 5 x c 5 Integrating sinnx and cosnx is relatively easy compared to integrating powers of sin and cos. ln sin x c nf '( x) cos f ( x) sin n 1 nf '( x) sin f ( x) cos nf '( x) sec f ( x) tan n 1 2 f ( x)dx dy ln cos ecx cot x c dx sin n f ( x) c f ( x)dx cosn f ( x) c n 1 f ( x)dx 30 x cos 3x 2 sin 3x 2 sin 5 3x 2 c tan n f ( x) c 4 Recognise that you have a function multiplied by its differential. A2 Differentiation and Integration Methods. 1 1 x2 1 sin 1 x c dx cos 1 x c dx 1 x2 1 1 x 2 dx dv u dx dx Notice similarity between sin and cos forms. tan 1 x c uv v Integration by parts. du dx dx Products of Trig Functions Integral sinn x cosm x dx, n is odd Method Factorise one sine out and convert the remaining sines to cosines using sin2 x 1cos2 x. Example sin x cos xdx sin x(1 cos x) cos sin x sin x cos x 3 2 2 4 1 cos x cos5 x c 5 sinn x cosm x dx, m is odd sinn x cosm x dx n and m are both odd sinn x cosm x dx n and m are both even. tann x secm x dx, n is odd. Factorise one cosine out and convert the remaining cosines to sines using cos2 x 1sin2 x. Use either 1. or 2. Use double angle formula for sine and/or half angle formulas to reduce the integral into a form that can be integrated. Factorise one tangent and one secant out and convert the remaining tangents to secants using tan2 xsec2 x 1. 5 2 x A2 Differentiation and Integration Methods. tann x secm x dx, m is even. Factorise two secants out and convert the remaining secants to tangents using sec2 x 1tan2 x . tan x sec xdx sec x(1 tan x) tan sec x sec tan x 3 4 2 2 2 2 5 1 tan x tan 6 x c 6 tann x secm x dx, n is odd and m is even. Use either 1. or 2. tann x secm x dx, n is even and m is odd. Each integral will be dealt with differently 6 3 x