If ₀^ 4 ^2 x 1+ x x ~d x= 1 a _ e ( a 3 )+ b 3 , where a , b N , then a + b is equal to_________

Step-by-step solution

aligned & ₀^ 2 ^2 x 1+ 1 2 2 x d x= ₀^ 4 1- 2 x 2+ 2 x d x & 1 2+ 2 x - 2 x 2+ 2 x aligned ( I ₁ ) - ( I ₂ ) aligned & ( I ₁ )= dx 2+ 2 x 1+ ^2 x & ₀^ 4 ^2 xdx 2 ^2 x +2 x+2 aligned aligned & x=t & 1 2 ₀^1 dt ( t + 1 2 )^2+ 3 4 = 6 3 & I ₂= ₀^ / 4 2 x 2+ 2 x dx = 1 2 ( 3 2 ) & I ₁- I ₂= 1 3 6 + 1 2 2 3 & a =2, ~b =6 aligned Ans. 8