Paragraph: Let I₁= ₀^ x 1+ ^2 x d x and I₂= ₀^ 1 1+ ^2 x d x Question: What is the value of I₁+I₂ I₁-I₂ ?

Step-by-step solution

I₁= ₀^ x 1+ ^2 x d x ....(1) I₁= ₀^ ( -x) 1+ ^2( -x) d x ....(2) (1) + (2), 2 I₁= ₀^ 1+ ^2 x d xI 1 d xI₁= 2 ₀^ ^2 x 2+ ^2 x d x[ f( -x)=f(x)]I₁= 2 2 ₀^ / 2 ^2 x d x 2+ ^2 x Let x=t, ^2 x d x=d tI₁= ₀^ d t t^2+( 2 )^2 I₁= . 2 ⁻¹ ( t 2 ) |₀ ^ = 2 ( 2 -0 )= ^2 2 2 Similarly, I₂= ₀^ 1 1+ ^2 x d x=2 ₀^ / 2 d x 1+ ^2 x =2 ₀^ / 2 d x 2 ^2 x+ ^2 x =2 ₀^ / 2 ^2 x d x 2 ^2 x+1 = ₀^ / 2 ^2 x d x ^2 x+ ( 1 2 )^2 Let x=t, ^2 x d x=d t ₀^ d t t^2+ ( 1 2 )^2 = . 2 ⁻¹ 2 t |₀ ^ = 2 ( 2 -0 )= 2 2 2 I₁+I₂ I₁-I₂ = ^2 2 2 + 2 2 2 ^2 2