html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cos

 http://functions.wolfram.com/01.07.21.2436.01

 Input Form

 Integrate[z^n Sin[b Sqrt[z] + d z + e] Cos[c z]^v, z] == (I (-1)^(n - 1) 2^(-2 - 2 n - v) (E^(2 I e) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^(-h + k) 4^k (I b)^(-h - k + 2 n) (I (b + 2 d Sqrt[z]))^(h + k) (-((I (b + 2 d Sqrt[z])^2)/d))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 d Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b + 2 d Sqrt[z])^2)/(4 d))] - 2 I d Sqrt[-((I (b + 2 d Sqrt[z])^2)/d)] Gamma[(1/2) (2 + h + k), -((I (b + 2 d Sqrt[z])^2)/(4 d))]), {k, 0, n}, {h, 0, k}] - E^((I b^2)/(2 d)) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^(-h + k) 4^k ((-I) b)^(-h - k + 2 n) ((-I) (b + 2 d Sqrt[z]))^ (h + k) ((I (b + 2 d Sqrt[z])^2)/d)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 d Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b + 2 d Sqrt[z])^2)/(4 d)] + 2 I d Sqrt[(I (b + 2 d Sqrt[z])^2)/d] Gamma[(1/2) (2 + h + k), (I (b + 2 d Sqrt[z])^2)/(4 d)]), {k, 0, n}, {h, 0, k}] + d^(2 + 2 n) E^((I b^2)/(4 d)) Sum[Binomial[v, s] ((-E^(I (2 e + b^2/(-4 d - 8 c s + 4 c v)))) (d + 2 c s - c v)^ (-2 n - 2) Sum[(-1)^(-h + k) 4^k (I b)^(-h - k + 2 n) (I (b + 2 (d + 2 c s - c v) Sqrt[z]))^(h + k) (-((I (b + 2 (d + 2 c s - c v) Sqrt[z])^2)/(d + 2 c s - c v)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-b) (b + 2 (d + 2 c s - c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b + 2 (d + 2 c s - c v) Sqrt[z])^2)/(4 (d + 2 c s - c v)))] + 2 I (d + 2 c s - c v) Sqrt[-((I (b + 2 (d + 2 c s - c v) Sqrt[z])^2)/(d + 2 c s - c v))] Gamma[(1/2) (2 + h + k), -((I (b + 2 (d + 2 c s - c v) Sqrt[z])^2)/(4 (d + 2 c s - c v)))]), {k, 0, n}, {h, 0, k}] - E^((I b^2)/(4 d + 8 c s - 4 c v)) (d + 2 c s - c v)^(-2 n - 2) Sum[(-1)^(-h + k) 4^k ((-I) b)^(-h - k + 2 n) ((-I) (b + 2 (d + 2 c s - c v) Sqrt[z]))^(h + k) ((I (b + 2 (d + 2 c s - c v) Sqrt[z])^2)/(d + 2 c s - c v))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 (d + 2 c s - c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b + 2 (d + 2 c s - c v) Sqrt[z])^2)/(4 (d + 2 c s - c v))] + 2 I (d + 2 c s - c v) Sqrt[(I (b + 2 (d + 2 c s - c v) Sqrt[z])^2)/(d + 2 c s - c v)] Gamma[(1/2) (2 + h + k), (I (b + 2 (d + 2 c s - c v) Sqrt[z])^ 2)/(4 (d + 2 c s - c v))]), {k, 0, n}, {h, 0, k}] + ((d - 2 c s + c v)^(-2 - 2 n) ((-E^(2 I e)) Sum[(-1)^(-h + k) 4^ k (I b)^(-h - k + 2 n) (I (b + 2 (d - 2 c s + c v) Sqrt[z]))^ (h + k) (-((I (b + 2 (d - 2 c s + c v) Sqrt[z])^2)/ (d - 2 c s + c v)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-b) (b + 2 (d - 2 c s + c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b + 2 (d - 2 c s + c v) Sqrt[z])^2)/(4 (d - 2 c s + c v)))] + 2 I (d - 2 c s + c v) Sqrt[-((I (b + 2 (d - 2 c s + c v) Sqrt[z])^2)/(d - 2 c s + c v))] Gamma[(1/2) (2 + h + k), -((I (b + 2 (d - 2 c s + c v) Sqrt[z])^2)/ (4 (d - 2 c s + c v)))]), {k, 0, n}, {h, 0, k}] + E^((I b^2)/(2 (d - 2 c s + c v))) Sum[(-1)^(-h + k) 4^k ((-I) b)^ (-h - k + 2 n) ((-I) (b + 2 (d - 2 c s + c v) Sqrt[z]))^ (h + k) ((I (b + 2 (d - 2 c s + c v) Sqrt[z])^2)/(d - 2 c s + c v))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ( (-b) (b + 2 (d - 2 c s + c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b + 2 (d - 2 c s + c v) Sqrt[z])^2)/ (4 (d - 2 c s + c v))] - 2 I (d - 2 c s + c v) Sqrt[(I (b + 2 (d - 2 c s + c v) Sqrt[z])^2)/(d - 2 c s + c v)] Gamma[(1/2) (2 + h + k), (I (b + 2 (d - 2 c s + c v) Sqrt[z])^2)/(4 (d - 2 c s + c v))]), {k, 0, n}, {h, 0, k}]))/E^((I b^2)/(4 (d - 2 c s + c v)))), {s, 0, Floor[(1/2) (-1 + v)]}]))/(d^(2 (1 + n)) E^((I (b^2 + 4 d e))/(4 d))) /; Element[n, Integers] && n >= 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sin ( z b + d z + e ) cos v ( c z ) z ( - 1 ) n - 1 2 - 2 n - v - 2 d - 2 ( n + 1 ) - ( b 2 + 4 d e ) 4 d ( b 2 4 d ( s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - b 2 4 d + 8 c s - 4 c v ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ) - h - k + 2 n ( - ( b + 2 ( d + 2 c s - c v ) z ) ) h + k ( ( b + 2 ( d + 2 c s - c v ) z ) 2 d + 2 c s - c v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( b + 2 ( d + 2 c s - c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 2 ( d + 2 c s - c v ) z ) 2 4 ( d + 2 c s - c v ) ) + 2 ( b + 2 ( d + 2 c s - c v ) z ) 2 d + 2 c s - c v ( d + 2 c s - c v ) Γ ( 1 2 ( h + k + 2 ) , ( b + 2 ( d + 2 c s - c v ) z ) 2 4 ( d + 2 c s - c v ) ) ) ) ( d + 2 c s - c v ) - 2 n - 2 - ( b 2 - 4 d - 8 c s + 4 c v + 2 e ) ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b ) - h - k + 2 n ( ( b + 2 ( d + 2 c s - c v ) z ) ) h + k ( - ( b + 2 ( d + 2 c s - c v ) z ) 2 d + 2 c s - c v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( d + 2 c s - c v ) - ( b + 2 ( d + 2 c s - c v ) z ) 2 d + 2 c s - c v Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 ( d + 2 c s - c v ) z ) 2 4 ( d + 2 c s - c v ) ) - b ( b + 2 ( d + 2 c s - c v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 ( d + 2 c s - c v ) z ) 2 4 ( d + 2 c s - c v ) ) ) ) ( d + 2 c s - c v ) - 2 n - 2 + - b 2 4 ( d - 2 c s + c v ) ( d - 2 c s + c v ) - 2 n - 2 ( b 2 2 ( d - 2 c s + c v ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ) - h - k + 2 n ( - ( b + 2 ( d - 2 c s + c v ) z ) ) h + k ( ( b + 2 ( d - 2 c s + c v ) z ) 2 d - 2 c s + c v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - b ( b + 2 ( d - 2 c s + c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 2 ( d - 2 c s + c v ) z ) 2 4 ( d - 2 c s + c v ) ) - 2 ( d - 2 c s + c v ) ( b + 2 ( d - 2 c s + c v ) z ) 2 d - 2 c s + c v Γ ( 1 2 ( h + k + 2 ) , ( b + 2 ( d - 2 c s + c v ) z ) 2 4 ( d - 2 c s + c v ) ) ) - 2 e k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b ) - h - k + 2 n ( ( b + 2 ( d - 2 c s + c v ) z ) ) h + k ( - ( b + 2 ( d - 2 c s + c v ) z ) 2 d - 2 c s + c v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( d - 2 c s + c v ) - ( b + 2 ( d - 2 c s + c v ) z ) 2 d - 2 c s + c v Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 ( d - 2 c s + c v ) z ) 2 4 ( d - 2 c s + c v ) ) - b ( b + 2 ( d - 2 c s + c v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 ( d - 2 c s + c v ) z ) 2 4 ( d - 2 c s + c v ) ) ) ) ) ) d 2 n + 2 - b 2 2 d ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ) - h - k + 2 n ( - ( b + 2 d z ) ) h + k ( ( b + 2 d z ) 2 d ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 2 d z ) 2 4 d ) + 2 ( b + 2 d z ) 2 d d Γ ( 1 2 ( h + k + 2 ) , ( b + 2 d z ) 2 4 d ) ) + 2 e ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b ) - h - k + 2 n ( ( b + 2 d z ) ) h + k ( - ( b + 2 d z ) 2 d ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 d z ) 2 4 d ) - 2 d - ( b + 2 d z ) 2 d Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 d z ) 2 4 d ) ) ) /; n v + Condition z z n z 1 2 b d z e c z v -1 n -1 2 -2 n -1 v -2 d -2 n 1 -1 b 2 4 d e 4 d -1 b 2 4 d -1 s 0 v -1 2 -1 Binomial v s -1 b 2 4 d 8 c s -1 4 c v -1 h 0 k k 0 n -1 k -1 h 4 k -1 b -1 h -1 k 2 n -1 b 2 d 2 c s -1 c v z 1 2 h k b 2 d 2 c s -1 c v z 1 2 2 d 2 c s -1 c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d 2 c s -1 c v z 1 2 Gamma 1 2 h k 1 b 2 d 2 c s -1 c v z 1 2 2 4 d 2 c s -1 c v -1 2 b 2 d 2 c s -1 c v z 1 2 2 d 2 c s -1 c v -1 1 2 d 2 c s -1 c v Gamma 1 2 h k 2 b 2 d 2 c s -1 c v z 1 2 2 4 d 2 c s -1 c v -1 d 2 c s -1 c v -2 n -2 -1 b 2 -4 d -1 8 c s 4 c v -1 2 e h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 d 2 c s -1 c v z 1 2 h k -1 b 2 d 2 c s -1 c v z 1 2 2 d 2 c s -1 c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k 2 d 2 c s -1 c v -1 b 2 d 2 c s -1 c v z 1 2 2 d 2 c s -1 c v -1 1 2 Gamma 1 2 h k 2 -1 b 2 d 2 c s -1 c v z 1 2 2 4 d 2 c s -1 c v -1 -1 b b 2 d 2 c s -1 c v z 1 2 Gamma 1 2 h k 1 -1 b 2 d 2 c s -1 c v z 1 2 2 4 d 2 c s -1 c v -1 d 2 c s -1 c v -2 n -2 -1 b 2 4 d -1 2 c s c v -1 d -1 2 c s c v -2 n -2 b 2 2 d -1 2 c s c v -1 h 0 k k 0 n -1 k -1 h 4 k -1 b -1 h -1 k 2 n -1 b 2 d -1 2 c s c v z 1 2 h k b 2 d -1 2 c s c v z 1 2 2 d -1 2 c s c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b b 2 d -1 2 c s c v z 1 2 Gamma 1 2 h k 1 b 2 d -1 2 c s c v z 1 2 2 4 d -1 2 c s c v -1 -1 2 d -1 2 c s c v b 2 d -1 2 c s c v z 1 2 2 d -1 2 c s c v -1 1 2 Gamma 1 2 h k 2 b 2 d -1 2 c s c v z 1 2 2 4 d -1 2 c s c v -1 -1 2 e h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 d -1 2 c s c v z 1 2 h k -1 b 2 d -1 2 c s c v z 1 2 2 d -1 2 c s c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k 2 d -1 2 c s c v -1 b 2 d -1 2 c s c v z 1 2 2 d -1 2 c s c v -1 1 2 Gamma 1 2 h k 2 -1 b 2 d -1 2 c s c v z 1 2 2 4 d -1 2 c s c v -1 -1 b b 2 d -1 2 c s c v z 1 2 Gamma 1 2 h k 1 -1 b 2 d -1 2 c s c v z 1 2 2 4 d -1 2 c s c v -1 d 2 n 2 -1 b 2 2 d -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 h 0 k k 0 n -1 k -1 h 4 k -1 b -1 h -1 k 2 n -1 b 2 d z 1 2 h k b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d z 1 2 Gamma 1 2 h k 1 b 2 d z 1 2 2 4 d -1 2 b 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h k 2 b 2 d z 1 2 2 4 d -1 2 e Binomial v v 2 -1 1 -1 \$CellContext`v 2 h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 d z 1 2 h k -1 b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d z 1 2 Gamma 1 2 h k 1 -1 b 2 d z 1 2 2 4 d -1 -1 2 d -1 b 2 d z 1 2 2 d -1 1 2 Gamma 1 2 h k 2 -1 b 2 d z 1 2 2 4 d -1 n v SuperPlus [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2002-12-18