Let F:[3,5] → R be a twice differentiable function on (3, 5) such that
F(x) = e⁻ˣ∫₃ˣ (3t² + 2t + 4F'(t))dt. If F'(4) = (α e^β - 224)/(e^β - 4)², then α + β is equal to _______________.

Solution

Maths sir · 4 steps

Do not differentiate first: integrate 4F'(t) straight away and F comes out as an explicit formula.

1. Find F(3). Put x = 3. The integral runs from 3 to 3, so it is zero, which gives F(3) = 0.
2. Integrate directly. Multiply both sides by eˣ. The term 4F'(t) integrates to 4F(x) - 4F(3) = 4F(x), and 3t² + 2t integrates to x³ + x² - 36. This gives eˣF(x) = x³ + x² - 36 + 4F(x).

   F(x) = (x³ + x² - 36)/(eˣ - 4)

3. Quotient rule. F'(x) = ((3x² + 2x)(eˣ - 4) - (x³ + x² - 36)eˣ)/(eˣ - 4)². At x = 4, 3x² + 2x = 56 and x³ + x² - 36 = 44, so the numerator is 56e⁴ - 224 - 44e⁴ = 12e⁴ - 224.
4. So. F'(4) = (12e⁴ - 224)/(e⁴ - 4)². Matching with the given form, α = 12 and β = 4, so α + β = 16.

The common slip: 60. Dropping the -(x³ + x² - 36)eˣ term of the quotient rule leaves 56e⁴ in the numerator, so α = 56 and the sum becomes 60.

The idea to keep. When F'(t) sits inside an integral of F, integrate it by the Fundamental Theorem before reaching for Leibniz. Always check the lower limit for a free value like F(3) = 0.

Answer from the NTA final answer key. Solution by PrepWiser.