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- Define Integration by Parts: We need to integrate the function with respect to from to . This is not a standard integral, so we will use integration by parts, which states that . We will let and .
- Choose and : First, we need to find by differentiating with respect to . Differentiating gives us .
- Find : Next, we need to find by integrating . We integrate with respect to . This is not a straightforward integration, and it seems like we might have chosen the wrong and for integration by parts. Let's reconsider our choice for and .
- Find : Let's choose and this time. This way, we can differentiate and hopefully get a simpler expression, and integrating should be straightforward.
- Apply Integration by Parts: We find by differentiating with respect to . Differentiating gives us .
- Reassess Strategy: Now, we integrate . Integrating is straightforward and gives us .
- Reassess Strategy: Now, we integrate . Integrating is straightforward and gives us .Now we apply the integration by parts formula: . Substituting the values we have:$\int(\(4\)\(-2\)x)e^{\(8\)x\(-2\)x^\(2\)} dx = (e^{\(8\)x\(-2\)x^\(2\)})(\(4\)x - x^\(2\)) - \int(\(4\)x - x^\(2\))(\(8\)\(-4\)x)e^{\(8\)x\(-2\)x^\(2\)} dx.
- Reassess Strategy: Now, we integrate \(dv\). Integrating \((4-2x) dx\) is straightforward and gives us \(v = 4x - x^2\).Now we apply the integration by parts formula: \(\int u dv = uv - \int v du\). Substituting the values we have:\(\newline\)\(\int(4-2x)e^{(8x-2x^2)} dx = (e^{(8x-2x^2)})(4x - x^2) - \int(4x - x^2)(8-4x)e^{(8x-2x^2)} dx\).We realize that the integral on the right side of the equation is more complicated than the original integral. This indicates that our choice for \(u\) and \(dv\) has not simplified the problem. We need to reassess our strategy for integrating this function.
