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

##### 1 Answer

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.

Now, I'm looking for a function and it's derivative. Here's where I notice that I have a second degree polynomial ( *x^2* will have a derivative that starts with an *x*. So:

If I decide that

then

but I don't have a

I'm allowed to manipulate coefficients in an integral (manipulating variables is trickier, and sometimes not possible). So I need to manipulate the *x* into being a *x*.

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 would really like to put a *x*, so that we can match the

=

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,

So for

For

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

Hope this helps.