4 ₀^1 ( 1 3+x^2 + 1+x^2 ) d x-3 _e( 3 ) is equal to :

Step-by-step solution

aligned & 4 ₀^1 1 3+ x ^2 + 1+ x ^2 dx -3 3 & =4 ₀^1 3+ x ^2 - 1+ x ^2 (3+ x ^2 )- (1- x ^2 ) dx - 3 2 3 & = & 2 [ x 2 3+ x ^2 + 3 2 ( x + 3+ x ^2 ) ₀^1 . & . - x 2 1+ x ^2 + 1 2 ( x + 1+ x ^2 ) ₀^1 ]- 3 2 3 aligned aligned = & 2 [ 1 2 4 + 3 2 (1+ 4 ) - 0+ 3 2 3 . & .- 1 2 2 + 1 2 (1+ 2 ) + 0+ 1 2 (0) ]- 3 2 3 aligned aligned & =2 [1+ 3 2 3- 3 4 3- 1 2 - 1 2 (1+ 2 ) ]- 3 2 3 & =2+3 3- 3 2 3- 2 - (1+ 2 )- 3 2 3 & =2- 2 - (1+ 2 ) aligned