If = _ 1 2 ^2 ⁻¹ x 2 x^2-3 x+2 d x then the value of 7 ( 2 7 ) is ____________. (Here, the inverse trigonometric function ⁻¹ x assumes values in (- 2 , 2 ) .)

Step-by-step solution

= _ 1 2 ^2 ⁻¹ x 2 x^2-3 x+2 d x ....(i) aligned & Let x= 1 t & d x=- 1 t^2 d t & = ₂^ 1 2 ⁻¹ ( 1 t ) 2 t^2 - 3 t +2 ( -1 t^2 ) d t aligned = _ 1 2 ^2 ⁻¹ t 2 t ^2-3 t +2 dt ....(ii) Now by (i) + (ii) aligned & 2 = _ 1 2 ^2 2 2 x^2-3 x+2 d x & = 8 _ 1 2 ^2 d x x^2- 3 x 2 +1 & = 8 _ 1 2 ^2 (x- 3 4 )^2+ 7 16 & = 8 7 4 [ ⁻¹ ( x- 3 4 7 4 ) ]_ 1 2 ^2 aligned aligned & = 2 7 [ ⁻¹ 4 x-3 7 ]_ 1 2 ^2 & = 2 7 [ ⁻¹ 5 7 - ⁻¹ (- 1 7 ) ] & = 2 7 ⁻¹ ( 5 7 + 1 7 ) 1- 5 7 & = 2 7 ⁻¹(3 7 ) aligned Now 7 ( 2 7 ) aligned & 7 ( ⁻¹(3 7 )