Cosine: Integration (formula 01.07.21.0706)

Cos

$Mathematica Notation$

Elementary Functions

Cos[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric functions

Involving powers of sin

Involving sin^m(b z^r+e) cos(c z^r+g)

http://functions.wolfram.com/01.07.21.0706.01

Input Form

Integrate[Sin[b z^r + e]^m Cos[c z^r + g], z] == -((1/r) (2^(-1 - m) z Binomial[m, m/2] ((E^(I g) Gamma[1/r, (-I) c z^r])/ ((-I) c z^r)^r^(-1) + Gamma[1/r, I c z^r]/(E^(I g) (I c z^r)^r^(-1))) (1 - Mod[m, 2]))) - (1/r) (2^(-1 - m) z Sum[(-1)^k Binomial[m, k] ((E^(I (-g + 2 e k - e m + (m Pi)/2)) Gamma[1/r, (-I) (-c + 2 b k - b m) z^r])/ ((-I) (-c + 2 b k - b m) z^r)^r^(-1) + Gamma[1/r, I (-c + 2 b k - b m) z^r]/ (E^(I (-g + 2 e k - e m + (m Pi)/2)) (I (-c + 2 b k - b m) z^r)^ r^(-1)) + (E^(I (g + 2 e k - e m + (m Pi)/2)) Gamma[1/r, (-I) (c + 2 b k - b m) z^r])/ ((-I) (c + 2 b k - b m) z^r)^r^(-1) + Gamma[1/r, I (c + 2 b k - b m) z^r]/ (E^(I (g + 2 e k - e m + (m Pi)/2)) (I (c + 2 b k - b m) z^r)^ r^(-1))), {k, 0, Floor[(1/2) (-1 + m)]}]) /; Element[m, Integers] && m > 0

Standard Form

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MathML Form

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<mi> b </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> / </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> g </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mrow> <mi> e </mi> 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Rule Form

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

2002-12-18