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Sigrid O'Kon

Feb 20, 2025

CALC HELP PLEASEEvaluate ∫(0 to 3) xf(x2)dx, if f is continuous and ∫(0 to 9) f(x)dx=4.?

I know this is kind of basic but I took this class last semester and cant remember. thanks
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Start off by letting:u = x^2 ==> du = 2x dx. Then. changing the bounds:Lower bound: x = 0 ==> u = 0^2 = 0Upper bound: x = 3 ==> u = 3^2 = 9. So, we have:∫ x*f(x^2) dx (from x=0 to 3)= 1/2 ∫ f(x^2) (2x dx) (from x=0 to 3)= 1/2 ∫ f(u) du (from u=0 to 9), by applying substitutions= 1/2 ∫ f(x) dx (from x=0 to 9), since x and u are both dummy variables= (1/2)(4), since ∫ f(x) dx (from x=0 to 9) = 4= 2.I hope this helps!... Show More
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Put u=x^2Then du=2xdxor xdx=(1/2)duWhen x=0, u=0; and when x=3, u=9So given integral is same as the integral from u=0 to u=9 of (1/2)f(u)duwhich is (1/2) integral of f(u)du=(1/2)(4)=2... Show More
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