The value of I = _ _e 2 ^ _e 3 x x^2 x^2+ ( _e 6-x^2 ) d x is

Step-by-step solution

Let I = _ 2 ^ 3 x x^2 x^2+ ( 6-x^2 ) d x array ll & Put x^2=t 2 x ~d x= dt & I = 1 2 _ ₂ ^ ₃ t t + ( 6- t ) dt ...(i) & I = 1 2 _ 2 ^ ₂ 3 ( 6- t ) ( 6- t )+ t dt ...(ii) array [ _a^b f(x) d x= _a^b f(a+b-x) d x ] Adding (i) and (ii), we get aligned & 2 I= 1 2 _ 2 ^ 3 dt = 1 2 ( 3- 2)= 1 2 ( 3 2 ) & I= 1 4 ( 3 2 ) aligned