AP DSC School Assistant Math Model Test 3

Concept:The integral can be recognized as the derivative of the function $\frac{x}{a+bx^2}$ with respect to $x$. This allows direct evaluation using the Fundamental Theorem of Calculus without substitution.Explanation:We are given: $$I = \int_{0}^{1} \frac{a - b x^2}{(a + b x^2)^2} \, dx.$$Notice that $$\frac{d}{dx} \left( \frac{x}{a + b x^2} \right) = \frac{ (a + b x^2) \cdot 1 - x \cdot (2b x) }{(a + b x^2)^2} = \frac{a + b x^2 - 2b x^2}{(a + b x^2)^2} = \frac{a - b x^2}{(a + b x^2)^2}.$$Thus the integrand is exactly the derivative of $\frac{x}{a + b x^2}$. Therefore, by the Fundamental Theorem of Calculus,$$I = \left[ \frac{x}{a + b x^2} \right]_{0}^{1} = \frac{1}{a + b \cdot 1^2} - \frac{0}{a + b \cdot 0^2} = \frac{1}{a + b} - 0 = \frac{1}{a + b}.$$No substitution or lengthy calculations are required once the derivative is identified.Answer:$\displaystyle \frac{1}{a+b}$, which corresponds to option D.