If a is a real number such that _ 0^a x d x a+4 then

Step-by-step solution

The given integral is ₀^ a x dx . Evaluating the definite integral: [ x^2 2 ]₀^ a = a^2 2 - 0 = a^2 2 The given inequality is a^2 2 a + 4 . Multiplying by 2: a^2 2a + 8 a^2 - 2a - 8 0 Factoring the quadratic expression: (a - 4)(a + 2) 0 The roots of the quadratic equation a^2 - 2a - 8 = 0 are a = 4 and a = -2 . For the inequality to hold, a must lie between the roots, inclusive: -2 a 4 . Answer: -2 a 4