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

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To integrate (\int x^3 \sin(x^2) , dx) using integration by parts, we need to choose (u) and (dv) such that when differentiated and integrated respectively, they simplify the integral or make it easier to solve.

Let's choose (u = x^3) and (dv = \sin(x^2) , dx). Then, (du = 3x^2 , dx) and (v = -\frac{1}{2}\cos(x^2)).

Now, we apply the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

Substituting our choices:

[ \int x^3 \sin(x^2) , dx = -\frac{1}{2}x^3\cos(x^2) - \int -\frac{1}{2}\cos(x^2) \cdot 3x^2 , dx ]

Simplify the integral:

[ = -\frac{1}{2}x^3\cos(x^2) + \frac{3}{2} \int x^2 \cos(x^2) , dx ]

Now, we have a new integral (\int x^2 \cos(x^2) , dx), which we can integrate by parts again if necessary.

The integration by parts process may need to be repeated until the integral becomes manageable or until it leads to a known integral form.

So, in summary, we integrate by parts to break down the original integral into simpler integrals that are easier to solve, and then repeat the process if necessary until we can evaluate the integral.

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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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