# How do you integrate #int x^3 e^x dx # using integration by parts?

Adding a constant to the solution,

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To integrate ∫x^3 * e^x dx using integration by parts:

Let u = x^3 and dv = e^x dx.

Then, differentiate u to get du and integrate dv to get v.

- Differentiate u: du = 3x^2 dx
- Integrate dv: v = e^x

Now, apply the integration by parts formula:

∫u dv = uv - ∫v du

Substitute the values of u, dv, du, and v into the formula:

∫x^3 * e^x dx = x^3 * e^x - ∫3x^2 * e^x dx

Now, the integral on the right side is similar to the original integral but with a lower power of x. You can apply integration by parts again if necessary, or you can directly integrate it. Continuing with the integration by parts method:

∫3x^2 * e^x dx

Let u = 3x^2 and dv = e^x dx.

Then, differentiate u to get du and integrate dv to get v.

- Differentiate u: du = 6x dx
- Integrate dv: v = e^x

Now, apply the integration by parts formula again:

∫u dv = uv - ∫v du

Substitute the values of u, dv, du, and v into the formula:

∫3x^2 * e^x dx = 3x^2 * e^x - ∫6x * e^x dx

Now, the integral on the right side is again similar to the original integral but with a lower power of x. You can apply integration by parts again if necessary, or you can directly integrate it.

Continuing this process until you reach an integral that you can easily solve, you can find the solution for ∫x^3 * e^x dx.

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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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