U substitution given (())() b a ò fgxg¢ xdx then the substitution u= gx( ) will convert this into the integral, (())() () bgb( ) aga òòfgxg¢ xdx= fudu. Product rule [ ]uv uv vu dx d = +′ 4. ©2005 be shapiro page 3 this document may not be reproduced, posted or published without permission.
Important Derivatives & Integrals
Quotient rule v2 vu uv v u dx d ′− ′ = 5.
What follows is a selection of entries.
Table of basic derivatives let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. Integral, (())() () bgb( ) aga òòfgxg¢ xdx= fudu. Morphing system mechanisms and controller architecture design teodor lucian grigorie, ruxandra mihaela botez, andrei vladimir popov école de technologie supérieure, montréal, québec h3c 1k3, canada mahmoud. Dx du (cu) c dx d = 3.
Integrattm ranges can be changed according to the rule:
The derivative of the sum or difference of two functions is equal to the sum or difference of the derivatives of the functions. Must know derivative and integral rules! For the following, let u and v be functions of x, let n be an integer, and let a, c, and c be constants. The copyright holder makes no representation about the accuracy, correctness, or
An indefinite integral computes the family of functions that are the antiderivative.
(a) the power rule : Sum and difference rule [ ]u v u v dx d ± = ±′ 3. Divide [ab,] into n subintervals of width ∆x and choose * x i from each interval. Dx du v dx dv (uv) u dx d = + 4.
Standard integration techniques note that all but the first one of these tend to be taught in a calculus ii class.
Standard integration techniques note that all but the first one of these tend to be taught in a calculus ii class. Derivatives & integrals jordan paschke derivatives here are a bunch of derivatives you should probably know. The tables give a nicer or more useful form of the answer than the one that the cas will yield. _f(e2) (x2 xl) if axk — x k x then the area ts:
Power rule [ ] u nu u dx d =n −1 ′ 7.
A definite integral is used to compute the area under the curve these are some of the most frequently encountered rules for differentiation and integration. (chain rule) if y = f(u) is differentiable on u = g(x) and u = g(x) is differentiable on point x, then the composite function y = f(g(x)) is differentiable and dx du du dy dx dy = 7. To) x (1) x0 = 1 (2) c0 = 0 (3) (cu)0 = c·u0 (4) (u±v) 0= u0 ±v (5) (uv) 0= u v +v0u (6) u v 0 = u0v −v0u v2 (7) (un) 0= nun−1u (a) 1 u 0 = − u0 u2 (b) (√ u)0 = u0 2 √ u Z sinm(x)cosn(x)dx = z sinm(x)cos2k+1(x)dx = z sinm(x)(cos2(x))k cos(x)dx = z sinm(x)(1sin2(x))k cos(x)dx
Power rule [ ]x =1 dx d 8.
Common derivatives and integrals derivative rules: D dx {un} = nu n−1. The integral can be calculated by ftnddtng the sum of each rectangle area: If y = cf(x) dx df cf x c dx d dx dy = ( ()) = example 1 if y = 8x, then = (8 ) =8 (x) =8(1) =8 dx d x dx d dx dy 3.
F0(x) x aax 1 sin(x) cos(x) cos(x) sin(x) tan(x) sec2(x) cot(x) csc2(x)
We highly recommend practicing with them (or creating ashcards for them) and looking at them occasionally until they are burned into your memory. Table 2.1, choose yp in the same line and determine its undetermined coefficients by substituting yp and its derivatives into (4). The derivative of the product of a constant and a function is equal to the constant times the derivative of the function. _f(e1) (xl a) second rectangle area is:
Then ( ) (*) 1 lim i b n a n i f x dx f x x →∞ = ∫ =∑ ∆.
Area = lim axk = f (x)dx and g — g(x) then: Derivative involving absolute value [ ]= ( )u′,u ≠0 u u u. Constant multiple rule [ ]cu cu dx d = ′, where c is a constant. Basic forms z xndx = 1 n+ 1 xn+1(1) z 1 x dx= lnjxj (2) z udv= uv z vdu (3) z 1 ax+ b dx= 1 a lnjax+ bj (4) integrals of rational functions z 1 (x+ a)2.
U substitution given (( )) ( ) b a ∫ f g x g x dx′ then the substitution u gx= ( ) will convert this into the integral, (( )) ( ) ( ) ( ) b gb( ) a ga ∫∫f g x g x dx f u du′ =.
Dx dv dx du (u v) dx d ± = ± 2. Differentiation formulas d dx k = 0 (1) d dx [f(x)±g(x)] = f0(x)±g0(x) (2) d dx [k ·f(x)] = k ·f0(x) (3) d dx [f(x)g(x)] = f(x)g0(x)+g(x)f0(x) (4) d dx f(x) g(x. Suppose f x( ) is continuous on [ab,]. U = u(x) and v = v(x) are differentiable functions of x;
If a term in your choice for yp happens to be a solution of the homogeneous ode corresponding to (4), multiply this term by x (or by x 2 if this solution corresponds to a double root of the
Calculus trigonometric derivatives and integrals strategy for evaluating r sinm(x)cosn(x)dx (a) if the power n of cosine is odd (n =2k +1), save one cosine factor and use cos2(x)=1sin2(x)to express the rest of the factors in terms of sine: Constant rule, [ ]c =0 dx d 6. If c ts a constant then: Integration by parts the standard formulas for integration by parts are.
F (x) ts deftned tn the range a to b and c ts a point tnstde this range then:
Dx dv wu dx du vw dx dw (uvw) uv dx d = + + 5. U0 = du dx is the derivative of u with respect to (w.r. U ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)−1/4.(3x2 + 4)d dx {u} = 12 u.u d dx { 2 − 4x2 + 7x5} = 1 2 2 − 4x2 + 7x5 (−8x + 35x4) d dx {c} = 0 , c is a constant ddx {6} = 0 , since ≅ 3.14 is a constant. C, n, and a > 0 are constants;
