Answers:
u = [tex]\boldsymbol{x^9+3}[/tex]
du = [tex]\boldsymbol{9x^8}[/tex] dx
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Explanation:
We use the derivative rule [tex]\frac{d}{dx}x^n = nx^{n-1}[/tex]
That allows us to go from [tex]u = x^9+3[/tex] to [tex]\frac{du}{dx} = 9x^8[/tex] which rearranges to [tex]du = 9x^8dx[/tex]
Notice the [tex]x^8dx[/tex] portion is found in the original integral. That extra '9' will have to move over to the other side.
So we'll need to have this
[tex]du = 9x^8dx \to x^8dx = \frac{du}{9}[/tex]
which means
[tex]\displaystyle \int x^8\sqrt{x^9+3} \ dx = \frac{1}{9}\int \sqrt{u} \ du[/tex]
