Begin2.DVI

∫ sinax dx sinax±cosax= x 2 ∓ln|sinax±cosax|+C ∫ cosax dx sinax±cosax=± x 2 + 1 2 aln|sinaxx±cosax|+C ∫ sinax dx α+β ...

∫ dx a^2 −b^2 cos^2 x=    1 a √ a^2 −b^2 tan−^1 ( a √ a^2 −b^2 tanx ) +C, a > b − 1 a√b^2 −a^2 tanh − 1 (√ a b^2 −a^ ...

∫ xtan−^1 xadx=^12 (x^2 +a^2 ) tan−^1 xa−a 2 ln|x^2 +a^2 |+C ∫ xcot−^1 xadx=^12 (x^2 +a^2 ) cot−^1 xa+a 2 x+C ∫ xsec ...

∫ 1 xcsc − 1 x adx=− ( a x+ 1 2 · 3 · 3 (x a ) 3 + 2 ·^14 ··^35 · 5 (x a ) 5 + 2 ·^14 ·· 63 ··^57 · 7 (x a ) 7 +··· ) +C ...

∫ eaxsinbx dx= ( asinbx−bcosbx a^2 +b^2 ) eax+C ∫ eaxcosbx dx= ( acosbx+bsinbx a^2 +b^2 ) eax+C ∫ eaxsinnbx dx= (asi ...

∫ eaxsin^2 bx dx= (a (^2) + 4b (^2) −a (^2) cos(2bx)− 2 absin(2bx) 2 a(a^2 + 4b^2 ) eax+C 569. ∫ eaxcos^2 bx dx= (a (^2) + 4 ...

∫ xsinhax dx=^1 axcoshax−a^12 sinhax+C ∫ x^2 sinhax dx= ( x^2 a + 2 a^3 ) coshax−^2 ax 2 sinhax+C ∫ xnsinhax dx=^1 a ...

∫ dx α+βsinhax= √^1 α^2 +β^2 ln ∣∣ ∣∣ ∣ βeax+α− √ α^2 +β^2 βeax+α+ √ α^2 +β^2 ∣∣ ∣∣ ∣+C ∫ dx (α+βsinhax)^2 = −β a(α^2 +β ...

∫ dx cosh^2 ax =^1 atanhax+C ∫ x dx cosh^2 ax= 1 axtanhax− 1 a^2 ln|coshax|+C ∫ dx coshnax= 1 (n−1)a xsinhax coshn−^ ...

∫ sinh^2 axcosh^2 ax dx= 321 asinh 4ax−^18 x+C ∫ sinhnaxcoshax dx=(n+ 1)^1 asinhn+1ax+C, n 6 =− 1 ∫ coshnaxsinhax dx ...

∫ dx sinh^2 axcoshax =−^1 atan−^1 (sinhax)−a^1 cschax+C ∫ dx sinh^2 axcosh^2 ax=− 2 acothax+C ∫ sinh (^2) ax coshaxd ...

∫ csch−^1 xadx=xcsch−^1 xa±asinh−^1 xa, +forx > 0 and−forx < 0 ∫ xsinh−^1 xadx= ( x^2 2 + a^2 4 ) sinh−^1 xa−^14 x ...

∫ xncoth−^1 xadx=n+ 1^1 xn+1coth−^1 xa−n+ 1a ∫ xn+1dx a^2 −x^2 ∫ xnsech−^1 xadx=    1 n+ 1x n+1sech− 1 x a+ a n+ ...

IfSm= ∫ xmsinnx dxandCm= ∫ xmcosnx dx, then Sm=−n^1 xmcosnx+mnCm− 1 and Cm=^1 nxmsinnx−mnSm− 1 IfI 1 = ∫ tanx dx, andIn= ∫ t ...

∫ J 1 (x) x dx=−J^1 (x) + ∫ J 0 (x)dx ∫ xνJν− 1 (x)dx=xνJν(x) +C ∫ x−νJν+1(x)dx=x−νJν(x) +C ∫ J 1 (x) xn dx= − 1 ...

Definite integrals General integration properties IfdFdx(x)=f(x), then ∫b a f(x)dx=F(x)|ba=F(b)−F(a) ∫∞ 0 f(x)dx= limb→∞ ∫b ...

Iff(x)is periodic with periodL, thenf(x+L) =f(x)for allxand ∫nL 0 f(x)dx=n ∫L 0 f(x)dx, for integer values ofn. ∫x 0 dx ∫x 0 ...

∫∞ 0 dx 1 −xn= π ncot π n ∫∞ 0 dx (a^2 x^2 +c^2 )(x^2 +b^2 )= π 2 bc 1 c+ab ∫∞ 0 dx (a^2 +x^2 )(b^2 +x^2 )= π 2 1 ab ...

∫π 0 sinpθsinqθ dθ= { 0 , p 6 =q π 2 , p=q ∫π 0 sinpθcosqθ dθ=    0 , p+qeven 2 p p^2 −q^2 , p+qodd ∫π 0 x dx a^2 ...

∫L −L cosmπxL sinnπxL dx= 0 for all integerm,nvalues ∫L −L cosmπxL cosnπxL dx=    0 , m 6 =n L 2 , m=n^6 = 0 L, m=n ...