If ₀^ x( x+1) e^ x+ x e^ x +1 d x=a+b ₀^ e^ x d x where a and b are positive rational numbers, then find the value of 100 (a^2+b^2 ) .

Step-by-step solution

Using King aligned & I= ₀^ x(1+ x) e^ x- x e^ - x +1 d x & I= ₀^ x(1+ x) e^ x e^ x +1 d x aligned ...(1) Add aligned & 2 I= ₀^ x(1+ x) e^ x (e^ x +1 ) e^ x +1 d x & I= 1 2 ₀^ x(1+ x) e^ x d x & I= 1 2 ₀^ e^ x ( x+1- ^2 x ) d x & I= 1 2 ₀^ e^ x d x+ 1 2 ₀^ e^ x ( x- ^2 x ) d x & I= 1 2 ₀^ e^ x d x+ 1 2 [e^ x (- x) ]₀^ & I= 1 2 ₀^ e^ x d x+ 1 2 [1+1] & I=1+ 1 2 ₀^ e^ x d x & 100 (1+ 1 4 )=125 & aligned