If xφ (x) = ∫₅ˣ (3t² - 2φ '(t))dt, x > -2, and φ(0) = 4, then φ(2) is __________.

Solution

Maths sir · 4 steps

Differentiate both sides to get rid of the integral, and the left side becomes an exact derivative.

1. Remove the integral. Differentiate both sides with respect to x. Use the product rule on the left and the fundamental theorem of calculus on the right: φ(x) + xφ'(x) = 3x² - 2φ'(x).
2. Spot the linear form. Collect the φ' terms: (x+2)φ' + φ = 3x². As a linear equation this is φ' + 1/(x+2)φ = 3x²/(x+2), with integrating factor e^(∫ dx/(x+2)) = x+2 because x > -2. Multiplying through, the left side is the derivative of (x+2)φ.

   d/dx[(x+2)φ(x)] = 3x² ⇒ (x+2)φ(x) = x³ + C

3. Find C. Put x = 0 and φ(0) = 4: 2 × 4 = 0 + C, so C = 8. Then φ(x) = (x³ + 8)/(x+2).
4. So. φ(2) = (8 + 8)/(2 + 2) = 16/4 = 4.

The common slip: 2. Dropping the constant of integration, so that (x+2)φ = x³, gives φ(2) = 8/4 = 2. The condition φ(0) = 4 is there to fix C = 8.

The idea to keep. When the unknown function sits inside an integral with variable limit x, differentiate first. Then check whether the left side is already a product-rule derivative before reaching for a formula.

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