# How do you evaluate the definite integral #int (8)/sqrt(3+4x)# from #[0,1]#?

definite integral

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To evaluate the definite integral (\int_{0}^{1} \frac{8}{\sqrt{3+4x}} , dx), we can use the substitution method.

Let (u = 3 + 4x). Then, (du = 4 , dx) and (dx = \frac{1}{4} du).

When (x = 0), (u = 3 + 4(0) = 3). When (x = 1), (u = 3 + 4(1) = 7).

Now, substituting into the integral:

[ \int_{0}^{1} \frac{8}{\sqrt{3+4x}} , dx = \int_{3}^{7} \frac{8}{\sqrt{u}} \cdot \frac{1}{4} , du ]

[ = 2 \int_{3}^{7} \frac{1}{\sqrt{u}} , du ]

[ = 2 \left[2\sqrt{u}\right]_{3}^{7} ]

[ = 4 \left(\sqrt{7} - \sqrt{3}\right) ]

So, (\int_{0}^{1} \frac{8}{\sqrt{3+4x}} , dx = 4(\sqrt{7} - \sqrt{3})).

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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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