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/01.07.21.2509.01

 Input Form

 Integrate[z^n Sin[b z^2 + d z]^m Cos[c z]^v, z] == 2^(-1 - m - v) ((1/(1 + n)) (2 z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2])) + 2 Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] (((-I) c (-2 s + v))^(-1 - n) Gamma[1 + n, (-I) c (-2 s + v) z] + (I c (-2 s + v))^(-1 - n) Gamma[1 + n, I c (-2 s + v) z]), {s, 0, Floor[(1/2) (-1 + v)]}] - (I/b^(2 n + 1)) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[((1/(2 k - m)^(2 n + 1)) (-1)^k Binomial[m, k] ((-E^(I m Pi)) (I b (-2 k + m))^n Sum[(1/d) (2^(-n + q) ((-I) d (2 k - m))^(1 + n - q) (d + 2 b z) (I (2 k - m) (d + 2 b z))^q (-((I (2 k - m) (d + 2 b z)^2)/b))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (2 k - m) (d + 2 b z)^2)/(4 b))]), {q, 0, n}] + E^((I d^2 (2 k - m))/(2 b)) (I b (2 k - m))^n Sum[(1/(d + 2 b z)) (2^(-n + q) b ((-I) d (-2 k + m))^(n - q) (I (-2 k + m) (d + 2 b z))^q ((I (2 k - m) (d + 2 b z)^2)/b)^ ((1 - q)/2) Binomial[n, q] Gamma[(1 + q)/2, (I (2 k - m) (d + 2 b z)^2)/(4 b)]), {q, 0, n}]))/ E^((I (d^2 (2 k - m) + 2 b m Pi))/(4 b)), {k, 0, Floor[(1/2) (-1 + m)]}] + (I/b^(2 n + 1)) Sum[(-1)^k Binomial[m, k] Sum[(1/(2 k - m)^(2 n + 1)) ((Binomial[v, s] (E^((I (-3 d^2 (-2 k + m)^2 + 4 b (2 k - m) m Pi + 2 c d (2 k - m) (2 s - v) - 3 c^2 (-2 s + v)^2))/(b (8 k - 4 m))) (I b (-2 k + m))^n Sum[2^(-n + q) (I (-2 d k + d m + 2 c s - c v))^(n - q) (I (2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z))^(1 + q) (-((I (2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z)^2)/ (b (2 k - m))))^((1/2) (-1 - q)) Binomial[n, q] Gamma[ (1 + q)/2, -((I (2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z)^2)/(b (8 k - 4 m)))], {q, 0, n}] - ((I b (2 k - m))^n Sum[2^(-n + q) (I (2 d k - d m - 2 c s + c v))^ (n - q) ((-I) (2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z))^(1 + q) ((I (2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z)^2)/(b (2 k - m)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z)^2)/(b (8 k - 4 m))], {q, 0, n}])/E^((I (d (-2 k + m) + c (-2 s + v))^2)/(b (8 k - 4 m))) + E^((I (-3 d^2 (-2 k + m)^2 + 4 b (2 k - m) m Pi - 2 c d (2 k - m) (2 s - v) - 3 c^2 (-2 s + v)^2))/(b (8 k - 4 m))) (I b (-2 k + m))^n Sum[2^(-n + q) ((-I) (2 d k - d m + 2 c s - c v))^(n - q) (I (2 d k - d m + 2 c s - c v + 4 b k z - 2 b m z))^(1 + q) (-((1/(b (2 k - m))) (I (d (-2 k + m) + c (-2 s + v) + 2 b (-2 k + m) z)^2)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (d (-2 k + m) + c (-2 s + v) + 2 b (-2 k + m) z)^2)/(b (8 k - 4 m)))], {q, 0, n}] - ((I b (2 k - m))^n Sum[2^(-n + q) (I (2 d k - d m + 2 c s - c v))^ (n - q) (I (d (-2 k + m) + c (-2 s + v) + 2 b (-2 k + m) z))^ (1 + q) ((1/(b (2 k - m))) (I (d (-2 k + m) + c (-2 s + v) + 2 b (-2 k + m) z)^2))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (d (-2 k + m) + c (-2 s + v) + 2 b (-2 k + m) z)^2)/(b (8 k - 4 m))], {q, 0, n}])/ E^((I (2 d k - d m - 2 c s + c v)^2)/(b (8 k - 4 m)))))/ E^((I ((-d^2) (-2 k + m)^2 + b (2 k - m) m Pi - c^2 (-2 s + v)^2))/ (b (4 k - 2 m)))), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}]) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sin m ( b z 2 + d z ) cos v ( c z ) z 2 - m - v - 1 ( 2 z n + 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) n + 1 + 2 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) 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]]]]] ( Γ ( n + 1 , - c ( v - 2 s ) z ) ( - c ( v - 2 s ) ) - n - 1 + ( c ( v - 2 s ) ) - n - 1 Γ ( n + 1 , c ( v - 2 s ) z ) ) - 1 b 2 n + 1 ( ( 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) k = 0 m - 1 2 1 ( 2 k - m ) 2 n + 1 ( ( - 1 ) k - ( ( 2 k - m ) d 2 + 2 b m π ) 4 b ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( d 2 ( 2 k - m ) 2 b ( b ( 2 k - m ) ) n q = 0 n 1 d + 2 b z ( 2 q - n b ( - d ( m - 2 k ) ) n - q ( ( m - 2 k ) ( d + 2 b z ) ) q ( ( 2 k - m ) ( d + 2 b z ) 2 b ) 1 - q 2 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( 2 k - m ) ( d + 2 b z ) 2 4 b ) ) - m π ( b ( m - 2 k ) ) n q = 0 n 1 d ( 2 q - n ( - d ( 2 k - m ) ) n - q + 1 ( d + 2 b z ) ( ( 2 k - m ) ( d + 2 b z ) ) q ( - ( 2 k - m ) ( d + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( 2 k - m ) ( d + 2 b z ) 2 4 b ) ) ) ) ) + 1 b 2 n + 1 ( k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 1 ( 2 k - m ) 2 n + 1 ( - ( - d 2 ( m - 2 k ) 2 - c 2 ( v - 2 s ) 2 + b ( 2 k - m ) m π ) b ( 4 k - 2 m ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - - ( 2 d k - d m - 2 c s + c v ) 2 b ( 8 k - 4 m ) ( q = 0 n 2 q - n ( ( 2 d k - d m + 2 c s - c v ) ) n - q ( ( d ( m - 2 k ) + 2 b z ( m - 2 k ) + c ( v - 2 s ) ) ) q + 1 ( ( d ( m - 2 k ) + 2 b z ( m - 2 k ) + c ( v - 2 s ) ) 2 b ( 2 k - m ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( d ( m - 2 k ) + 2 b z ( m - 2 k ) + c ( v - 2 s ) ) 2 b ( 8 k - 4 m ) ) ) ( b ( 2 k - m ) ) n - - ( d ( m - 2 k ) + c ( v - 2 s ) ) 2 b ( 8 k - 4 m ) ( q = 0 n 2 q - n ( ( 2 d k - d m - 2 c s + c v ) ) n - q ( - ( 2 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) ) q + 1 ( ( 2 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) 2 b ( 2 k - m ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( 2 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) 2 b ( 8 k - 4 m ) ) ) ( b ( 2 k - m ) ) n + ( - 3 d 2 ( m - 2 k ) 2 - 3 c 2 ( v - 2 s ) 2 - 2 c d ( 2 k - m ) ( 2 s - v ) + 4 b ( 2 k - m ) m π ) b ( 8 k - 4 m ) ( b ( m - 2 k ) ) n q = 0 n 2 q - n ( - ( 2 d k - d m + 2 c s - c v ) ) n - q ( ( 2 d k + 4 b z k - d m + 2 c s - c v - 2 b m z ) ) q + 1 ( - ( d ( m - 2 k ) + 2 b z ( m - 2 k ) + c ( v - 2 s ) ) 2 b ( 2 k - m ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( d ( m - 2 k ) + 2 b z ( m - 2 k ) + c ( v - 2 s ) ) 2 b ( 8 k - 4 m ) ) + ( - 3 d 2 ( m - 2 k ) 2 - 3 c 2 ( v - 2 s ) 2 + 2 c d ( 2 k - m ) ( 2 s - v ) + 4 b ( 2 k - m ) m π ) b ( 8 k - 4 m ) ( b ( m - 2 k ) ) n q = 0 n 2 q - n ( ( - 2 d k + d m + 2 c s - c v ) ) n - q ( ( 2 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) ) q + 1 ( - ( 2 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) 2 b ( 2 k - m ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( 2 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) 2 b ( 8 k - 4 m ) ) ) ) ) ) /; n m + v + Condition z z n b z 2 d z m c z v 2 -1 m -1 v -1 2 z n 1 Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 2 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s Gamma n 1 -1 c v -1 2 s z -1 c v -1 2 s -1 n -1 c v -1 2 s -1 n -1 Gamma n 1 c v -1 2 s z -1 1 b 2 n 1 -1 Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 1 2 k -1 m 2 n 1 -1 -1 k -1 2 k -1 m d 2 2 b m 4 b -1 Binomial m k d 2 2 k -1 m 2 b -1 b 2 k -1 m n q 0 n 1 d 2 b z -1 2 q -1 n b -1 d m -1 2 k n -1 q m -1 2 k d 2 b z q 2 k -1 m d 2 b z 2 b -1 1 -1 q 2 -1 Binomial n q Gamma q 1 2 -1 2 k -1 m d 2 b z 2 4 b -1 -1 m b m -1 2 k n q 0 n 1 d -1 2 q -1 n -1 d 2 k -1 m n -1 q 1 d 2 b z 2 k -1 m d 2 b z q -1 2 k -1 m d 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 k -1 m d 2 b z 2 4 b -1 1 b 2 n 1 -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 1 2 k -1 m 2 n 1 -1 -1 -1 d 2 m -1 2 k 2 -1 c 2 v -1 2 s 2 b 2 k -1 m m b 4 k -1 2 m -1 Binomial v s -1 -1 2 d k -1 d m -1 2 c s c v 2 b 8 k -1 4 m -1 q 0 n 2 q -1 n 2 d k -1 d m 2 c s -1 c v n -1 q d m -1 2 k 2 b z m -1 2 k c v -1 2 s q 1 d m -1 2 k 2 b z m -1 2 k c v -1 2 s 2 b 2 k -1 m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d m -1 2 k 2 b z m -1 2 k c v -1 2 s 2 b 8 k -1 4 m -1 b 2 k -1 m n -1 -1 d m -1 2 k c v -1 2 s 2 b 8 k -1 4 m -1 q 0 n 2 q -1 n 2 d k -1 d m -1 2 c s c v n -1 q -1 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z q 1 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z 2 b 2 k -1 m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z 2 b 8 k -1 4 m -1 b 2 k -1 m n -3 d 2 m -1 2 k 2 -1 3 c 2 v -1 2 s 2 -1 2 c d 2 k -1 m 2 s -1 v 4 b 2 k -1 m m b 8 k -1 4 m -1 b m -1 2 k n q 0 n 2 q -1 n -1 2 d k -1 d m 2 c s -1 c v n -1 q 2 d k 4 b z k -1 d m 2 c s -1 c v -1 2 b m z q 1 -1 d m -1 2 k 2 b z m -1 2 k c v -1 2 s 2 b 2 k -1 m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d m -1 2 k 2 b z m -1 2 k c v -1 2 s 2 b 8 k -1 4 m -1 -3 d 2 m -1 2 k 2 -1 3 c 2 v -1 2 s 2 2 c d 2 k -1 m 2 s -1 v 4 b 2 k -1 m m b 8 k -1 4 m -1 b m -1 2 k n q 0 n 2 q -1 n -2 d k d m 2 c s -1 c v n -1 q 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z q 1 -1 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z 2 b 2 k -1 m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z 2 b 8 k -1 4 m -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18