If ₀^ 1 / 2 ⁻¹ ( 1 2 ( 1+x x - x 1+x ) ) d x=p (2+ 3 )- q , then p q is equal to:

Step-by-step solution

aligned & ₀^ 1 / 2 ⁻¹ ( 1 2 ( 1+x x - x 1+x ) ) d x= ₀^ 1 / 2 ⁻¹ ( 1 2 ( 1 x(1+x) ) ) d x & = ₀^ 1 / 2 ⁻¹ ( 1 4 x^2+4 x+1-1 ) d x= ₀^ 1 / 2 ⁻¹ ( 1 (2 x+1)^2-1 ) d x aligned Put 2 x+1= d x= 2 d aligned I & = ₀^ 3 ⁻¹( ) 2 d = ₀^ 3 ( 2 - ) 2 d I & = 4 ( )₀^ / 3 - 1 2 ( - ( + ))₀^ / 3 & = 4 (2-1)- 1 2 ( 2 2- (2+ 3 ) )= - 12 + (2+ 3 ) 2 p (2+ 3 )- q aligned p q=6