# How do you integrate #(t^2)e^(4t)#?

For the integral, use integration by parts:

This can be entered into our earlier computations:

Include the integration constant:

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To integrate (t^2)e^(4t), you can use integration by parts. Let u = t^2 and dv = e^(4t) dt. Then, differentiate u to get du = 2t dt and integrate dv to get v = (1/4)e^(4t). Now, apply the integration by parts formula: ∫ u dv = uv - ∫ v du. Substituting the values, we get: ∫ (t^2)e^(4t) dt = (t^2)(1/4)e^(4t) - ∫ (1/4)e^(4t) * 2t dt. Simplifying, we have: ∫ (t^2)e^(4t) dt = (1/4)t^2e^(4t) - (1/2)∫ te^(4t) dt. Now, we need to integrate te^(4t). Again, we use integration by parts with u = t and dv = e^(4t) dt. Differentiating u, we get du = dt and integrating dv, we obtain v = (1/4)e^(4t). Applying the formula, we have: ∫ te^(4t) dt = (1/4)te^(4t) - (1/4)∫ e^(4t) dt. Simplifying, we get: ∫ te^(4t) dt = (1/4)te^(4t) - (1/16)e^(4t). Finally, substituting this result back into the earlier equation, we get: ∫ (t^2)e^(4t) dt = (1/4)t^2e^(4t) - (1/2)((1/4)te^(4t) - (1/16)e^(4t)) + C, where C is the constant of integration. Simplifying further, we obtain the final result: ∫ (t^2)e^(4t) dt = (1/4)t^2e^(4t) - (1/8)te^(4t) + (1/32)e^(4t) + C.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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