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Homework Statement
If [tex]f[/tex] is continuous and [tex]\int^{9}_{0}f(x)dx = 4[/tex], find [tex]\int^{3}_{0}xf(x^{2})dx[/tex]
Homework Equations
None required
The Attempt at a Solution
Don't really know where to begin, but I tried:
for [tex]\int^{3}_{0}xf(x^{2})dx[/tex]
let:
[tex]u = x^{2}[/tex]
[tex]du = 2xdx[/tex]
substitute
[tex]\int^{3}_{0}xf(x^{2})dx = \int^{9}_{0}\frac{1}{2}f(u)du
[/tex]
NOW, for [tex]\int^{9}_{0}f(x)dx = 4[/tex]
let
[tex]u = x[/tex]
[tex]du = dx[/tex]
then we have
[tex]\int^{9}_{0}f(x)dx = 4 = \int^{9}_{0}f(u)du[/tex]
so now...
[tex]\int^{9}_{0}\frac{1}{2}f(u)du = (1/2)(4) = 2[/tex]
so our result, the answer to the second integral is 2.
But I'm pretty sure I'm wrong, I didn't really know where else to go with this so that's what I tried. This question is in the "u-substitution" section of our text, so that's probably the method we use, some sort of substitution.
Any help is appreciated, thank you in advance!