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\int x \sin x d x |
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Answer» Let u = x, which implies du/dx = 1 and let dv/dx = sin(x). Integrating this to get v gives v =–cos(x). So our integral is now of the form required for integration by parts. ∫ x sin(x)dx = ∫ u(dv/dx) dx = uv –∫ v(du/dx)dx =–xcos(x)– ∫–cos(x)*1dx =–xcos(x)– ∫–cos(x) dx =–xcos(x)+ ∫cos(x)dx The integral of cos(x) is equal to sin(x). We can check this by differentiating sin(x), which does indeed give cos(x). Finally, as with all integration without limits, there must be a constant added, which I'll callc. So the final answer is ∫ x sin(x)dx =–xcos(x)+ sin(x) + c |
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