Saturday, September 12, 2015

calculus - Evaluating: int3xsinleft(fracx4right),dx.




3xsin(x4)dx



4xxcosx44cos(x4)4xcos(x4)dx3(4cos(x4)+4xcos(x4)dx)4cos(x4)xdx uvvdudxdx



4cos(x4)+4cosx/4xdx



3[4cos(x4)+4(cos(x4)ln(x))]



12cos(x/4)+12cos(x/4)ln(x) wrong.







3xsin(x4)dx
cos(x/4)x1xcos(x/4)dx



3[4cos(x/4)+4xsin(x/4)]



12cos(x/4)+48xsin(x/4)





In the text above is my work done to solve the following question:





Find the indefinite integral of: [3xsin(x4)]





The bordered area is the furthest I got (there should be a 3 at the front to multiply the whole equation but I usually remember to add that at the end) The part where I wrote "wrong" is what I thought the answer was, I assumed it would be such. What I'm having troubles with is integrating the cos(x/4)x. Would I need to make it cos(x/4)x and then integrate by parts to get that integral? Thanks in advance, I hope I made some sense in what I'm trying to achieve. I guess what I'm looking for, is a way to integrate cos(x/4)x


Answer




I think you have made a mistake in the application of integration by parts. Taking u as x and dvdx as sinx4, we should get
3xsinx4dx=3(4xcosx44cosx4dx)=12xcosx4+48sinx4+C.


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