Let J = _ -3 ³ 9 - x^2 (9 + x^2) 81 + x^4 1 1 + 5^x dx . The value of 24 2 J is equal to ______.

Step-by-step solution

Using the property _ -a ^ a f(x) 1+c^x dx = ₀^ a f(x) dx for an even function f(x) , we have: J = ₀³ 9 - x^2 (9 + x^2) 81 + x^4 dx Dividing the numerator and denominator by x^2 : J = ₀³ 9 x^2 - 1 (x + 9 x ) x^2 + 81 x^2 dx J = ₀³ 9 x^2 - 1 (x + 9 x ) (x + 9 x )^2 - 18 dx Let t = x + 9 x . Then dt = (1 - 9 x^2 ) dx ( 9 x^2 - 1 ) dx = -dt . When x 0 , t . When x = 3 , t = 3 + 3 = 6 . J = _ ⁶ -dt t t^2 - 18 = ₆^ dt t t^2 - 18 Using the standard integral dx x x^2 - a^2 = 1 a ⁻¹ ( x a ) + C : J = [ 1 3 2 ⁻¹ ( t 3 2 ) ]₆