# How do you integrate #int sin^3x# by integration by parts method?

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To integrate ( \int \sin^3(x) , dx ) using integration by parts, you can use the following steps:

- Start by rewriting ( \sin^3(x) ) as ( \sin^2(x) \cdot \sin(x) ).
- Choose ( u = \sin^2(x) ) and ( dv = \sin(x) , dx ).
- Calculate ( du ) and ( v ) by taking derivatives and integrals, respectively.
- Apply the integration by parts formula: ( \int u , dv = uv - \int v , du ).
- Substitute the values of ( u ), ( du ), ( v ), and ( dv ) into the formula.
- Evaluate the resulting integral.

Following these steps, you'll find that ( \int \sin^3(x) , dx = -\frac{1}{3}\cos(x) + \frac{1}{3}\sin^3(x) + C ), where ( C ) is the constant of integration.

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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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