Handbook of mathematics for engineers and scienteists part 170 ppsx

T2.2. TABLES OF DEFINITE INTEGRALS 1151 2.  1 0 x n e –ax dx = n! a n+1 – e –a n  k=0 n! k! 1 a n–k+1 , a > 0, n = 1, 2, 3.  ∞ 0 x n e –ax dx = n! a n+1 , a > 0, n = 1, 2, 4.  ∞ 0 e –ax √ x dx =  π a , a > 0. 5.  ∞ 0 x ν–1 e –μx dx = Γ(ν) μ ν , μ, ν > 0. 6.  ∞ 0 dx 1 + e ax = ln 2 a . 7.  ∞ 0 x 2n–1 dx e px – 1 =(–1) n–1  2π p  2n B 2n 4n , n = 1, 2, ; the B m are Bernoulli numbers. 8.  ∞ 0 x 2n–1 dx e px + 1 =(1 – 2 1–2n )  2π p  2n |B 2n | 4n , n = 1, 2, 9.  ∞ –∞ e –px dx 1 + e –qx = π q sin(πp/q) , q > p > 0 or 0 > p > q. 10.  ∞ 0 e ax + e –ax e bx + e –bx dx = π 2b cos  πa 2b  , b > a. 11.  ∞ 0 e –px – e –qx 1 – e –(p+q)x dx = π p + q cot πp p + q , p, q > 0. 12.  ∞ 0  1 – e –βx  ν e –μx dx = 1 β B  μ β , ν + 1  . 13.  ∞ 0 exp  –ax 2  dx = 1 2  π a , a > 0. 14.  ∞ 0 x 2n+1 exp  –ax 2  dx = n! 2a n+1 , a > 0, n = 1, 2, 15.  ∞ 0 x 2n exp  –ax 2  dx = 1×3× × (2n – 1) √ π 2 n+1 a n+1/2 , a > 0, n = 1, 2, 16.  ∞ –∞ exp  –a 2 x 2 bx  dx = √ π |a| exp  b 2 4a 2  . 17.  ∞ 0 exp  –ax 2 – b x 2  dx = 1 2  π a exp  –2 √ ab  , a, b > 0. 18.  ∞ 0 exp  –x a  dx = 1 a Γ  1 a  , a > 0. 1152 INTEGRALS T2.2.3. Integrals Involving Hyperbolic Functions 1.  ∞ 0 dx cosh ax = π 2|a| . 2.  ∞ 0 dx a + b cosh x = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2 √ b 2 – a 2 arctan √ b 2 – a 2 a + b if |b| > |a|, 1 √ a 2 – b 2 ln a + b + √ a 2 – b 2 a + b – √ a 2 + b 2 if |b| < |a|. 3.  ∞ 0 x 2n dx cosh ax =  π 2a  2n+1 |E 2n |, a > 0;theB m are Bernoulli numbers. 4.  ∞ 0 x 2n cosh 2 ax dx = π 2n (2 2n – 2) a(2a) 2n |B 2n |, a > 0. 5.  ∞ 0 cosh ax cosh bx dx = π 2b cos  πa 2b  , b > |a|. 6.  ∞ 0 x 2n cosh ax cosh bx dx = π 2b d 2n da 2n 1 cos  1 2 πa/b  , b > |a|, n = 1, 2, 7.  ∞ 0 cosh ax cosh bx cosh(cx) dx = π c cos  πa 2c  cos  πb 2c  cos  πa c  +cos  πb c  , c > |a| + |b|. 8.  ∞ 0 xdx sinh ax = π 2 2a 2 , a > 0. 9.  ∞ 0 dx a + b sinh x = 1 √ a 2 + b 2 ln a + b + √ a 2 + b 2 a + b – √ a 2 + b 2 , ab ≠ 0. 10.  ∞ 0 sinh ax sinh bx dx = π 2b tan  πa 2b  , b > |a|. 11.  ∞ 0 x 2n sinh ax sinh bx dx = π 2b d 2n dx 2n tan  πa 2b  , b > |a|, n = 1, 2, 12.  ∞ 0 x 2n sinh 2 ax dx = π 2n a 2n+1 |B 2n |, a > 0. T2.2.4. Integrals Involving Logarithmic Functions 1.  1 0 x a–1 ln n xdx=(–1) n n! a –n–1 , a > 0, n = 1, 2, 2.  1 0 ln x x + 1 dx =– π 2 12 . 3.  1 0 x n ln x x + 1 dx =(–1) n+1  π 2 12 + n  k=1 (–1) k k 2  , n = 1, 2, T2.2. TABLES OF DEFINITE INTEGRALS 1153 4.  1 0 x μ–1 ln x x + a dx = πa μ–1 sin(πμ)  ln a – π cot(πμ)  , 0 < μ < 1. 5.  1 0 |ln x| μ dx = Γ(μ + 1), μ >–1. 6.  ∞ 0 x μ–1 ln(1 + ax) dx = π μa μ sin(πμ) ,–1 < μ < 0. 7.  1 0 x 2n–1 ln(1 + x) dx = 1 2n 2n  k=1 (–1) k–1 k , n = 1, 2, 8.  1 0 x 2n ln(1 + x) dx = 1 2n + 1  ln 4 + 2n+1  k=1 (–1) k k  , n = 0, 1, 9.  1 0 x n–1/2 ln(1 + x) dx = 2 ln 2 2n + 1 + 4(–1) n 2n + 1  π – n  k=0 (–1) k 2k + 1  , n = 1, 2, 10.  ∞ 0 ln a 2 + x 2 b 2 + x 2 dx = π(a – b), a, b > 0. 11.  ∞ 0 x p–1 ln x 1 + x q dx =– π 2 cos(πp/q) q 2 sin 2 (πp/q) , 0 < p < q. 12.  ∞ 0 e –μx ln xdx =– 1 μ (C +lnμ), μ > 0, C = 0.5772 T2.2.5. Integrals Involving Trigonometric Functions T2.2.5-1. Integrals over a finite interval. 1.  π/2 0 cos 2n xdx= π 2 1×3× × (2n – 1) 2×4× × (2n) , n = 1, 2, 2.  π/2 0 cos 2n+1 xdx= 2×4× × (2n) 1×3× × (2n + 1) , n = 1, 2, 3.  π/2 0 x cos n xdx=– m–1  k=0 (n – 2k + 1)(n – 2k + 3) (n – 1) (n – 2k)(n – 2k + 2) n 1 n – 2k + ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ π 2 (2m – 2)!! (2m – 1)!! if n = 2m – 1, π 2 8 (2m – 1)!! (2m)!! if n = 2m, m = 1, 2, 4.  π 0 dx (a+b cos x) n+1 = π 2 n (a+b) n √ a 2 –b 2 n  k=0 (2n–2k–1)!! (2k–1)!! (n–k)! k!  a+b a–b  k , a>|b|. 5.  π/2 0 sin 2n xdx= π 2 1×3× × (2n – 1) 2×4× × (2n) , n = 1, 2, 1154 INTEGRALS 6.  π/2 0 sin 2n+1 xdx= 2×4× × (2n) 1×3× × (2n + 1) , n = 1, 2, 7.  π 0 x sin μ xdx= π 2 2 μ+1 Γ(μ + 1)  Γ  μ + 1 2  2 , μ >–1. 8.  π/2 0 sin xdx √ 1 – k 2 sin 2 x = 1 2k ln 1 + k 1 – k . 9.  π/2 0 sin 2n+1 x cos 2m+1 xdx= n! m! 2(n + m + 1)! , n, m = 1, 2, 10.  π/2 0 sin p–1 x cos q–1 xdx= 1 2 B  1 2 p, 1 2 q  . 11.  2π 0 (a sin x + b cos x) 2n dx = 2π (2n – 1)!! (2n)!!  a 2 + b 2  n , n = 1, 2, 12.  π 0 sin xdx √ a 2 + 1 – 2a cos x =  2 if 0 ≤ a ≤ 1, 2/a if 1 < a. 13.  π/2 0 (tan x) λ dx = π 2 cos  1 2 πλ  , |λ| < 1. T2.2.5-2. Integrals over an infinite interval. 1.  ∞ 0 cos ax √ x dx =  π 2a , a > 0. 2.  ∞ 0 cos ax –cosbx x dx =ln    b a    , ab ≠ 0. 3.  ∞ 0 cos ax –cosbx x 2 dx = 1 2 π(b – a), a, b ≥ 0. 4.  ∞ 0 x μ–1 cos ax dx = a –μ Γ(μ)cos  1 2 πμ  , a > 0, 0 < μ < 1. 5.  ∞ 0 cos ax b 2 + x 2 dx = π 2b e –ab , a, b > 0. 6.  ∞ 0 cos ax b 4 + x 4 dx = π √ 2 4b 3 exp  – ab √ 2  cos  ab √ 2  +sin  ab √ 2  , a, b > 0. 7.  ∞ 0 cos ax (b 2 + x 2 ) 2 dx = π 4b 3 (1 + ab)e –ab , a, b > 0. 8.  ∞ 0 cos ax dx (b 2 + x 2 )(c 2 + x 2 ) = π  be –ac – ce –ab  2bc  b 2 – c 2  , a, b, c > 0. 9.  ∞ 0 cos  ax 2  dx = 1 2  π 2a , a > 0. REFERENCES FOR CHAPTER T2 1155 10.  ∞ 0 cos  ax p  dx = Γ(1/p) pa 1/p cos π 2p , a > 0, p > 1. 11.  ∞ 0 sin ax x dx = π 2 sign a. 12.  ∞ 0 sin 2 ax x 2 dx = π 2 |a|. 13.  ∞ 0 sin ax √ x dx =  π 2a , a > 0. 14.  ∞ 0 x μ–1 sin ax dx = a –μ Γ(μ)sin  1 2 πμ  , a > 0, 0 < μ < 1. 15.  ∞ 0 sin  ax 2  dx = 1 2  π 2a , a > 0. 16.  ∞ 0 sin  ax p  dx = Γ(1/p) pa 1/p sin π 2p , a > 0, p > 1. 17.  ∞ 0 sin x cos ax x dx = ⎧ ⎨ ⎩ π 2 if |a| < 1, π 4 if |a| = 1, 0 if 1 < |a|. 18.  ∞ 0 tan ax x dx = π 2 sign a. 19.  ∞ 0 e –ax sin bx dx = b a 2 + b 2 , a > 0. 20.  ∞ 0 e –ax cos bx dx = a a 2 + b 2 , a > 0. 21.  ∞ 0 exp  –ax 2  cos bx dx = 1 2  π a exp  – b 2 4a  . 22.  ∞ 0 cos(ax 2 )cosbx dx =  π 8a  cos  b 2 4a  +sin  b 2 4a  , a, b > 0. 23.  ∞ 0 (cos ax +sinax)cos(b 2 x 2 ) dx = 1 b  π 8 exp  – a 2 2b  , a, b > 0. 24.  ∞ 0  cos ax +sinax  sin(b 2 x 2 ) dx = 1 b  π 8 exp  – a 2 2b  , a, b > 0. References for Chapter T2 Bronshtein, I. N. and Semendyayev, K. A., Handbook of Mathematics, 4th Edition, Springer-Verlag, Berlin, 2004. Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961. Gradshteyn,I.S.andRyzhik,I.M.,Tables of Integrals, Series, and Products, 6th Edition, Academic Press, New York, 2000. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 1, Elementary Functions, Gordon & Breach, New York, 1986. Prudnikov,A.P.,Brychkov,Yu.A.,andMarichev,O.I.,Integrals and Series, Vol. 2, Special Functions, Gordon & Breach, New York, 1986. Prudnikov, A. P.,Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 3, More Special Functions, Gordon & Breach, New York, 1988. Zwillinger,D., CRC Standard Mathematical Tables and Formulae, 31st Edition, CRC Press, Boca Raton, 2002. Chapter T3 Integral Transforms T3.1. Tables of Laplace Transforms T3.1.1. General Formulas No. Original function, f(x) Laplace transform,  f(p)=  ∞ 0 e –px f(x) dx 1 af 1 (x)+bf 2 (x) a  f 1 (p)+b  f 2 (p) 2 f(x/a), a > 0 a  f(ap) 3  0 if 0 < x < a, f(x – a)ifa < x e –ap  f (p) 4 x n f(x); n = 1, 2, (–1) n d n dp n  f (p) 5 1 x f(x)  ∞ p  f(q) dq 6 e ax f(x)  f (p – a) 7 sinh(ax)f(x) 1 2   f(p – a)–  f(p + a)  8 cosh(ax)f(x) 1 2   f(p – a)+  f(p + a)  9 sin(ωx)f(x) – i 2   f (p – iω)–  f (p + iω)  , i 2 =–1 10 cos(ωx)f(x) 1 2   f(p – iω)+  f(p + iω)  , i 2 =–1 11 f(x 2 ) 1 √ π  ∞ 0 exp  – p 2 4t 2   f(t 2 ) dt 12 x a–1 f  1 x  , a >–1  ∞ 0 (t/p) a/2 J a  2 √ pt   f(t) dt 13 f(a sinh x), a > 0  ∞ 0 J p (at)  f(t) dt 14 f(x + a)=f(x) (periodic function) 1 1 – e ap  a 0 f(x)e –px dx 15 f(x + a)=–f(x) (antiperiodic function) 1 1 + e –ap  a 0 f(x)e –px dx 16 f  x (x) p  f (p)–f(+0) 17 f (n) x (x) p n  f(p)– n  k=1 p n–k f (k–1) x (+0) 1157 . > 0. References for Chapter T2 Bronshtein, I. N. and Semendyayev, K. A., Handbook of Mathematics, 4th Edition, Springer-Verlag, Berlin, 2004. Dwight, H. B., Tables of Integrals and Other Mathematical. 1988. Zwillinger,D., CRC Standard Mathematical Tables and Formulae, 31st Edition, CRC Press, Boca Raton, 2002. Chapter T3 Integral Transforms T3.1. Tables of Laplace Transforms T3.1.1. General Formulas No 1961. Gradshteyn,I.S.andRyzhik,I.M.,Tables of Integrals, Series, and Products, 6th Edition, Academic Press, New York, 2000. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series,