# How do you find the integral #int_0^(2)(10x)/sqrt(3-x^2)dx# ?

This is a classic case if what's called u-substitution. Meaning that you have to find a function ( u) and it's derivative (du) in the expression. Both the function and it's derivative may be hidden behind coefficients and confusing notation.

In this case, I might start by using exponents to rewrite the integral without fractions.

If I decide that

then

We start with:

We can pull a coefficient factor out of the integral entirely. We don't have to, but I find it makes for a more clear picture.

Now we have our u and our du. But there's one other thing we need to be aware of.

This is a definite integral, and the 0 and the 2 represent x values, not u values. But we have a way to convert them,

Now we substitute all our x's for u's.

Hope this helps.

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To find the integral ( \int_{0}^{2} \frac{10x}{\sqrt{3-x^2}} , dx), you can use the substitution method:

Let ( u = 3 - x^2 ).

Then, ( du = -2x , dx ) and ( x , dx = -\frac{1}{2} , du ).

When ( x = 0 ), ( u = 3 - 0^2 = 3 ).

When ( x = 2 ), ( u = 3 - 2^2 = -1 ).

Now, we rewrite the integral with respect to ( u ):

( \int_{3}^{-1} \frac{-10}{2} \frac{1}{\sqrt{u}} , du ).

( = -5 \int_{3}^{-1} u^{-\frac{1}{2}} , du ).

( = -5 \left[ \frac{u^{\frac{1}{2}}}{\frac{1}{2}} \right]_{3}^{-1} ).

( = -10 \left[ \sqrt{u} \right]_{3}^{-1} ).

( = -10 \left( \sqrt{-1} - \sqrt{3} \right) ).

( = -10 \left( i\sqrt{1} - \sqrt{3} \right) ).

( = -10i + 10\sqrt{3} ).

Therefore, the integral ( \int_{0}^{2} \frac{10x}{\sqrt{3-x^2}} , dx = -10i + 10\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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