How do you integrate #int x^3cos(x^2)# using integration by parts?

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Answer 1 · 2016-08-28T13:39:40

#= 1/2 x^2 sin(x^2) + 1/2cos(x^2) + C#

#int x^3cos(x^2)#

#int x^2 . x cos(x^2)#

#= int x^2 d/dx (1/2 sin(x^2))#

ready for IBP
#= 1/2 x^2 sin(x^2) - 1/2 int d/dx (x^2) sin(x^2)#

#= 1/2 x^2 sin(x^2) - int x sin(x^2)#

#= 1/2 x^2 sin(x^2) - int d/dx ( - 1/2cos(x^2))#

#= 1/2 x^2 sin(x^2) + 1/2cos(x^2) + C#