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.
Now, I'm looking for a function and it's derivative. Here's where I notice that I have a second degree polynomial (
If I decide that
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
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
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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