If _a^b(x^2 - x) , dx = 18, _a^b x^3 , dx = 0 , then a + b is equal to

Step-by-step solution

Given _a^b x^3 , dx = 0 [ x^4 4 ]_a^b = 0 b^4 - a^4 4 = 0 b^4 = a^4 b = a or b = -a If b = a , then _a^b (x^2 - x) , dx = 0 , which contradicts the given condition _a^b (x^2 - x) , dx = 18 . Therefore, b = -a . This directly gives a + b = 0 . To find the exact values, substituting b = -a into the first integral: _a^ -a (x^2 - x) , dx = 18 - _ -a ^a (x^2 - x) , dx = 18 Using properties of definite integrals for even and odd functions: - ( 2 ₀^a x^2 , dx - 0 ) = 18 -2 [ x^3 3 ]₀^a = 18 - 2a^3 3 = 18 a^3 = -27 a = -3