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i) ∫01xe2xcos⁡3(x) dx \int_{0}^{1} x e^{2x} \cos^{3}(x) \, dx

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Evaluate definite integrals using the chain ruleright chevron icon

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Q. i) ∫01xe2xcos⁡3(x) dx \int_{0}^{1} x e^{2x} \cos^{3}(x) \, dx
  1. Given Integral: We are given the integral to evaluate: \newline∫01x⋅e2x⋅cos⁡(3x) dx\int_{0}^{1}x\cdot e^{2x}\cdot \cos(3x)\,dx\newlineThis integral is quite complex due to the presence of an exponential function, a trigonometric function, and a polynomial term. A common method to tackle such integrals is integration by parts, which is based on the formula:\newline∫u dv=uv−∫v du\int u\, dv = uv - \int v\, du\newlineHowever, due to the complexity of the functions involved, we might need to apply integration by parts multiple times or use a different strategy. In this case, the integral does not have a straightforward antiderivative, and it is likely that a numerical method or series expansion would be more appropriate to evaluate it. Since the problem asks for an exact, simplified answer, we must acknowledge that we cannot provide one using elementary functions.

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