F(x) and g(x) are any continuous functions; 1.1 motivation the primary issue in differentiation is the noise corruption. Derivatives derivative applications limits integrals integral applications integral approximation series ode multivariable calculus laplace transform taylor/maclaurin series fourier series.
derivative and integral table MHS Physics Semester 2
Common derivatives and integrals provided by the academic center for excellence 1 reviewed june 2008 common derivatives and integrals derivative rules:
Quotient rule v2 vu uv v u dx d ′− ′ =
Then ( ) (*) 1 lim i b n a n i f x dx f x x →∞ = ∫ =∑ ∆. Area under a curve 3. So to integrate a function f(x), you write ∫ f (x)dx D dx {un} = nu n−1.
C, n, and a > 0 are constants (1) z (f(x)+g(x))dx = z f(x)dx+ z g(x)dx (2) z cf(x)dx = c z f(x)dx (3) z un du = un+1 n+1 +c, n 6= −1 (a) z 1 u du = z du u = ln|u|+c (b) z 1 √ u du = z du √ u = 2 √ u+c (c) z du = u+c (4) z e udu = e +c (5) z
Let us discuss here the general formulas used in. Differentiation and integration rules a derivative computes the instantaneous rate of change of a function at different values. A definite integral is used to compute the area under the curve It is well known that a pure differentiator is not physically realizable due to its noise amplification property.
Divide [ab,] into n subintervals of width ∆x and choose * x i from each interval.
Indefinite integral :∫f x dx f x c( ) = +( ) V2 dx dv u dx du v v u dx d − = 6. 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: ∫∫ ∫ ∫udv uv vdu=−= udv uv vdu− choose uand then compute and dv duby differentiating uand compute vby using the fact that v dv=∫.
Suppose f x( ) is continuous on [ab,].
(a) the power rule : Product rule [ ]uv uv vu dx d = +′ 4. 9781133105060_app_g.qxp 12/27/11 1:47 pm page g1 appendix g.1 differentiation and integration formulas g1 g formulas g.1 differentiation and integration formulas use differentiation and integration tables to supplement differentiation and integration techniques. Table of basic derivatives let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists.
Du = du dx dx = u0 dx;
The integration of a function f (a) is given by f (a) and is written as. (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. The improvement of the differentiation and integration techniques is the key in enhancing the pid performance. Differentiation is used to calculate the gradient of a.
Table of integrals basic forms (1)!xndx= 1 n+1 xn+1 (2) 1 x!dx=lnx (3)!udv=uv!vdu (4) u(x)v!(x)dx=u(x)v(x)#v(x)u!(x)dx rational functions (5) 1 ax+b!dx= 1 a ln(ax+b) (6) 1 (x+a)2!dx= 1 x+a (7)!(x+a)ndx=(x+a)n a 1+n + x 1+n #$ % &', n!1 (8)!x(x+a)ndx= (x+a)1+n(nx+xa) (n+2)(n+1) (9) dx!1+x2 =tan1x (10) dx!a2+x2 = 1 a tan1(x/a) (11) xdx!a2+x2.
Integration formulas z dx = x+c (1) z xn dx = xn+1 n+1 +c (2) z dx x = ln|x|+c (3) z ex dx = ex +c (4) z ax dx = 1 lna ax +c (5) z lnxdx = xlnx−x+c (6) z sinxdx = −cosx+c (7) z cosxdx = sinx+c (8) z tanxdx = −ln|cosx|+c (9) z cotxdx = ln|sinx|+c (10) z secxdx = ln|secx+tanx|+c (11) z cscxdx = −ln |x+cot +c (12) z sec2 xdx = tanx+c (13) z csc2 xdx = −cotx+c (14) z Differentiation (cu) ' — cu' (c constm1t) integration u'vdx(byparts) uv —cos x + c sin x + c in eos + c in + c in + + c in — cot xl + c — amtan — arcsin — arcs inh — arccosh — + c — åsin2r+c tan x — x + c —cot x— x + c du sin x tan sec dr esc dr sin2 x dr dr tan2 x dr cot2 x dr du dy (chain rule) ae —sin cosh x sinh x Consider the signal (or (a) use the differentiation and integration properties. An indefinite integral computes the family of functions that are the antiderivative.
Basic differentiation rules basic integration formulas derivatives and integrals © houghton mifflin company, inc.
Line equations functions arithmetic & comp. Integration differentiation is two different parts of calculus that deal with the changes. Because the derivative of a constant function is always zero, so the differentiation process eliminates the ‘c’. Dx du (cu) c dx d = 3.
F (a) is called primitive, f (a) is called the integrand and c is constant of integration, a is variable.
'( ) 3 ( ) '( ) 3 ( ) '( ) 3 ( ) 2 2 2 h x x f x g x x f x f x x f x = = = = = = although integration has been introduced as an antiderivative, the symbol for integration is ‘∫’. Dx dv dx du (u v) dx d ± = ± 2. ∫ ∫ f (a) da = f (a) +c, where right hand side shows integral of f (a) with respect to a. U = u(x) is differentiable function of x;
The standard formulas for integration by parts are, bb b aa a.
Constant multiple rule [ ]cu cu dx d = ′, where c is a constant. Dx dv wu dx du vw dx dw (uvw) uv dx d = + + 5. 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 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.
Differentiation formulas d d d 1.
Dx du v dx dv (uv) u dx d = + 4.
