# How do you integrate #{x/(sqrt(4+4x^2))} dx# from 0 to 2?

The answer is

Therefore,

so,

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To integrate ( \frac{x}{\sqrt{4+4x^2}} ) with respect to (x) from 0 to 2, you can use the substitution method. Let ( u = 4 + 4x^2 ). Then, ( du = 8x , dx ). Rearranging, we get ( \frac{1}{8} du = x , dx ). Substituting (u) and (du), the integral becomes ( \int \frac{1}{8} \frac{1}{\sqrt{u}} , du ) from 4 to 20. This simplifies to ( \frac{1}{8} \int u^{-1/2} , du ) from 4 to 20. Integrating ( u^{-1/2} ) yields ( 2u^{1/2} ). Evaluating this from 4 to 20 and then multiplying by ( \frac{1}{8} ) gives ( \frac{1}{8} (2\sqrt{20} - 2\sqrt{4}) ), which simplifies to ( \frac{1}{4} (\sqrt{20} - 2) ). Hence, the definite integral from 0 to 2 is ( \frac{1}{4} (\sqrt{20} - 2) ).

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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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