CALC HELP PLEASEEvaluate ∫(0 to 3) xf(x2)dx, if f is continuous and ∫(0 to 9) f(x)dx=4.?
I would appreciate some help with a calculus problem. I need to evaluate the integral ∫(0 to 3) x f(x²) dx, given that f is a continuous function and that the integral ∫(0 to 9) f(x) dx equals 4. I realize this may seem basic, but I took this class last semester and am having trouble recalling the details. Thank you!
2 Answers
Start off by letting:
u = x^2 ==> du = 2x dx.
Then. changing the bounds:
Lower bound: x = 0 ==> u = 0^2 = 0
Upper 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!
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