Define the real-valued function f on the set of real numbers by f(x) = INTEGRATION(from 0 to 1)?

(x^2 + t^2)/(2 - t)dt


Consider the curve y = f(x). It represents


A) a straight line;


B) a parabola;


C) a hyparabola;


D) an ellipse;


Kindly explain your answer...

Define the real-valued function f on the set of real numbers by f(x) = INTEGRATION(from 0 to 1)?
let w = 2 - t ... dw = - dt





∫[2 , 1] (x^2 + (2-w)^2)/ w [-dw]


=∫[1,2] (x^2 + 4 - 4w + w^2)/w dw


= ∫[1,2] [(x^2+4)/w - 4 + w] dw


= (x^2+4) lnw - 4w + w^2/2 ... from 1 to 2


= (x^2+4) ln2 - 8 + 2 - (x^2+4) ln1 + 4 - 1/2





thus


y = ln2(x^2+4) - 5/2





thus this is a parabola... x is quadratic and y is linear.








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