How do you evaluate the definite integral #int 2(pi)x(cos^(-1)(x))dx# from 0 to 1?
Working first with the unbounded integral, we should apply integration by parts. Let:
Then:
Simplifying:
Plugging this into our previous expression:
Now applying the bounds, the original integral equals:
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To evaluate the definite integral ( \int_{0}^{1} 2\pi x \cos^{-1}(x) , dx ), you can use integration by parts, where ( u = \cos^{-1}(x) ) and ( dv = 2\pi x , dx ). Then, you differentiate ( u ) to find ( du ), and integrate ( dv ) to find ( v ). Finally, apply the integration by parts formula:
[ \int u , dv = uv - \int v , du ]
After evaluating the integral, substitute the limits of integration (0) and (1), and then calculate the difference to find the value of the definite 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.
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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