Friday, April 21, 2017

calculus - show that intinfty0fracsin3(x)x3dx=frac3pi8



show that


0sin3(x)x3dx=3π8


using different ways


thanks for all


Answer



Let f(y)=0sin3yxx3dx Then, f(y)=30sin2yxcosyxx2dx=340cosyxcos3yxx2dx f(y)=340sinyx+3sin3yxxdx Therefore, f(y)=940sin3yxxdx340sinyxxdx


Now, it is quite easy to prove that 0sinaxxdx=π2signuma


Therefore, f(y)=9π8signumy3π8signumy=3π4signumy Then, f(y)=3π4|y|+C Note that, f(0)=0, therefore, C=0. f(y)=3π8y2signumy+D Again, f(0)=0, therefore, D=0.


Hence, f(1)=0sin3xx3=3π8


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