3∫xsin(x4)dx
−4xxcosx4⇒−4cos(x4)−∫−4xcos(x4)dx3(−4cos(x4)+∫4xcos(x4)dx)∫4cos(x4)xdx uv−∫vdudxdx
−4cos(x4)+4∫cosx/4xdx
3[−4cos(x4)+4(cos(x4)ln(x))]
−12cos(x/4)+12cos(x/4)ln(x)→ wrong.
∫3xsin(x4)dx
∫cos(x/4)x→1x∫cos(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
3∫xsinx4dx=3(−4xcosx4−∫−4cosx4dx)=−12xcosx4+48sinx4+C.
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