・区間[0,∞]での定積分                      GAI 氏

 次の10問に挑戦願う。(ただし結果は明示的な式で示して下さい。)

[1]∫[x=0,∞] sin(x)/x dx

[2]∫[x=0,∞] sin(2*x)/x dx

[3]∫[x=0,∞] sin(3*x)/x dx

[4]∫[x=0,∞] (sin(x)/x)^2 dx

[5]∫[x=0,∞] (sin(x)/x)^3 dx

[6]∫[x=0,∞] sin^3(x)/x dx

[7]∫[x=0,∞] sin^3(x)/x^2 dx

[8]∫[x=0,∞] sin^5(x)/x dx

[9]∫[x=0,∞] (sin(2*x)/x)^3 dx

[10]∫[x=0,∞] (sin(3*x)/x)^3 dx


(コメント) WolframAlphaさんにお助けを乞うと、

[1] π/2  [2] π/2  [3] π/2  [4] π/2  [5] 3π/8  [6] π/4  [7] (3log3)/4

[8] 3π/16  [9] 3π/2  [10] 27π/8

となるらしいです。



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