How do you evaluate the definite integral #int (2t1)^2 # from #[0,1]#?
Transforming the limits:
Integral becomes:
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To evaluate the definite integral (\int_{0}^{1} (2t1)^2 dt):

Expand the integrand: ((2t1)^2 = (2t1)(2t1)).

Expand the squared binomial: ((2t1)^2 = 4t^2  4t + 1).

Integrate each term: (\int_{0}^{1} 4t^2  4t + 1 dt).

Use the power rule for integration: (\int t^n dt = \frac{t^{n+1}}{n+1} + C).

Integrate each term separately: (\int_{0}^{1} 4t^2 dt  \int_{0}^{1} 4t dt + \int_{0}^{1} 1 dt).

Evaluate each integral: (\left[\frac{4t^3}{3}\right]{0}^{1}  \left[2t^2\right]{0}^{1} + \left[t\right]_{0}^{1}).

Substitute the upper and lower limits and compute the values: (\left(\frac{4}{3}\right)  \left(2  0\right) + \left(1  0\right)).

Simplify the expression: (\frac{4}{3}  2 + 1).

Combine like terms: (\frac{4}{3}  1).

Calculate the final result: (\frac{1}{3}).
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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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