# How do you integrate #int x^3 e^(x^2 ) dx # using integration by parts?

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To integrate ∫x^3 * e^(x^2) dx using integration by parts, we apply the integration by parts formula:

∫u dv = uv - ∫v du

Let u = x^3 and dv = e^(x^2) dx. Then, differentiate u to get du and integrate dv to get v.

du = 3x^2 dx v = ∫e^(x^2) dx

To integrate v, we perform a substitution. Let w = x^2, then dw = 2x dx. Therefore, ∫e^(x^2) dx becomes ∫(1/2)e^w dw, which integrates to (1/2)e^(x^2).

Now, we can apply the integration by parts formula:

∫x^3 * e^(x^2) dx = x^3 * (1/2)e^(x^2) - ∫(1/2)e^(x^2) * 3x^2 dx

Simplify:

= (1/2)x^3 * e^(x^2) - (3/2)∫x^2 * e^(x^2) dx

We can apply integration by parts again to evaluate the remaining integral. Let u = x^2 and dv = e^(x^2) dx. Then, differentiate u to get du and integrate dv to get v.

du = 2x dx v = (1/2)e^(x^2)

Now, we apply the integration by parts formula:

= (1/2)x^3 * e^(x^2) - (3/2)[x^2 * (1/2)e^(x^2) - ∫(1/2)e^(x^2) * 2x dx]

= (1/2)x^3 * e^(x^2) - (3/2)[(1/2)x^2 * e^(x^2) - ∫x * e^(x^2) dx]

We continue this process until we can integrate the remaining term. However, integrating xe^(x^2) requires another substitution.

Let z = x^2, then dz = 2x dx. Therefore, ∫xe^(x^2) dx becomes ∫(1/2)e^z dz, which integrates to (1/2)e^(x^2).

Substitute back into the equation and simplify to obtain the final result.

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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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