# How do you evaluate the definite integral #intsec^2(t/4) # from #[0, pi]#?

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To evaluate the definite integral ( \int_{0}^{\pi} \sec^2(\frac{t}{4}) , dt ) over the interval ([0, \pi]), you can use a substitution. Let ( u = \frac{t}{4} ), so ( du = \frac{1}{4} dt ). Then the integral becomes ( 4\int_{0}^{\pi/4} \sec^2(u) , du ).

The integral of ( \sec^2(u) ) is ( \tan(u) ). So, integrating ( \sec^2(u) ) from (0) to (\pi/4) gives ( \tan(\pi/4) - \tan(0) = 1 - 0 = 1 ).

Thus, the value of the definite integral ( \int_{0}^{\pi} \sec^2(\frac{t}{4}) , dt ) is ( 4 \times 1 = 4 ).

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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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