Paragraph: Let 2 x^2-1 x^2+1 d x= U (x) V (x)-3 U (x)+ V (x) +c Question: What is | U ^2(x)- V ^2(x) | equal to?

Step-by-step solution

aligned & I=2 x^2-1 x^2+1 d x=2 ( x^2+1 x^2+1 - 2 x^2+1 ) d x & =2 ( x^2+1 - 2 x^2+1 ) d x & =2 [ x 2 x^2+1 + 1 2 (x+ x^2+1 )-2 (x+ x^2+1 )+c . & =x x^2+1 + (x+ x^2+1 )-4 (x+ x^2+1 ) & =x x^2+1 -3 (x+ x^2+1 )+c & =U(x) V(x)-3 U(x)+V(x) +c & U(x)=x and V(x)= x^2+1 aligned aligned & Now, | U ^2( x )- V ^2( x ) | & = | x ^2- ( x ^2+1 ) |= | x ^2- x ^2-1 |=|-1|=1 aligned