List of integrals of exponential functions
List of integrals of exponential functions 2 where where and is the Gamma Function when , , and when , , and Definite integrals for, which is the logarithmic mean
87 Table of Integrals
to integrate each of the following functions with respect to t a) et, b) e5t, c) t7, d) √ t, e) cos5t, f) e−t Answers 1 a) x2 2 + c, b) x7 7 + c, c) x−1 −1 + c = −x−1 + c, or −1 x + c, d) x−2 −2 + c = −1 2 x−2 + c, or − 1 2x2 +c, e) lnx+c, f) x3/2 3/2 +c = 2 3 x3/2 +c, g) x1/2 1/2 +c = 2x1/2 +c, h) 1 3 e3x +c, i) 1
Table of Integrals
e t 2 dt (51) xe xdx= (x 1)e (52) Z xeaxdx= x a 1 2 eax (53) Z x2exdx= x2 2x+ 2 ex (54) Z x2 eaxdx= x a 2x a2 + 2 a3 (55) Z 3exdx= 3 2 + 6 6 ex (56) Z xn eax d= x eax a n Z 1 (57) Z xneax dx= ( n1) an+1 [1 + n; ax]; where ( a;x) = Z 1 x ta 1e t dt (58) Z eax 2 dx= i p ˇ 2 p a erf ix p a (59) Z e ax 2 dx= p ˇ 2 p a erf x p a (60) Z xe ax 2 dx
Table of Integrals - University of Alberta
Table of Integrals Z sinaxsinbxdx = sin[(a−b)x] 2(a−b) − sin[(a+b)x] 2(a+b), (a26= b2) Z sin2axdx = x 2 − sin2ax 4a Z cosaxcosbxdx = sin[(a−b)x] 2(a−b
Table of Basic Integrals Basic Forms
(20) Z x p x (adx= 8
Table of Integrals
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Intégrales doubles [Correction]
exp(−(x2 + y2))dxdyetg(R) = ZZ B R exp(−(x2 + y2))dxdy a)Montrerqueg(R) 6 f(R) 6 g(R √ 2) b)Endéduirelavaleurde Z +∞ 0 e−t2dt Exercice 24 [ 02546 ] [Correction] SoitC(R) lequartdedisquex> 0,y> 0,x2 + y2 6 R2,R>0 a)Montrerque Z R 0 e−t2 dt 2 estcomprisentre ZZ C(R) e−x2−y2 dxdyet ZZ C(R √ 2) e−x2−y2 dxdy b)Calculer ZZ C
3 Contour integrals and Cauchy’s Theorem
curve given by r(t), a t b, then we can view r0(t) as a complex-valued curve, and then Z C f(z)dz= Z b a f(r(t)) r0(t)dt; where the indicated multiplication is multiplication of complex numbers (and not the dot product) Another notation which is frequently used is the following We denote a parametrized curve in the complex plane by z(t),
4 Improper Integrals - homeiitmacin
t R t f(x) dx= 0 for every t2R, but the integrals R c 1 f(x) dxand 1 c f(x) dxdo not exist for any c2R Next we consider integrals of functions de ned over in nite integrals of the form (a;1) and (1 ;b) De nition 4 2 (i) Suppose f is de ned on (a;1) and R 1 t f(x)dxexists for all t>a If lim ta Z 1 t f(x) dxexists, then we de ne the improper
Tempered distributions and the Fourier transform
18 1 TEMPERED DISTRIBUTIONS AND THE FOURIER TRANSFORM by (1 36) O K( )(˚) = Z K˚ dxdy: Theorem 1 2 There is a 1-1 correspondence between continuous linear oper-
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