How do you integrate #int (sin^1x)^2# using integration by parts?
We have:
So, we have:
So we see that:
Or:
Then:
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To integrate ( \int (\sin^{1} x)^2 ) using integration by parts, follow these steps:
 Let ( u = (\sin^{1} x)^2 ) and ( dv = dx ).
 Compute ( du ) and ( v ) using differentiation and integration, respectively.
 Apply the integration by parts formula: ( \int u , dv = uv  \int v , du ).
Here are the detailed steps:

Let ( u = (\sin^{1} x)^2 ). Then, ( du = 2 \sin^{1} x \cdot \frac{1}{\sqrt{1x^2}} , dx ).

Let ( dv = dx ). Then, ( v = x ).

Apply the integration by parts formula: [ \int (\sin^{1} x)^2 , dx = uv  \int v , du ] [ = x(\sin^{1} x)^2  \int x \cdot 2 \sin^{1} x \cdot \frac{1}{\sqrt{1x^2}} , dx ]
Now, the integral ( \int x \cdot 2 \sin^{1} x \cdot \frac{1}{\sqrt{1x^2}} , dx ) can be tackled using substitution or other methods to simplify further if needed.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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