Let the domain of the function f( x )= ₂ ₄ ₆ (3+4 x-x^2 ) be ( a , ~b ) . If ₀^ b - a [ x ^2 ] dx = p - q - r , p , q , r N , gcd ( p , q , r )=1, where [ ] is the greatest integer function, then p + q + r is equal to

Step-by-step solution

aligned & ₄ ₆ (3+4 x-x^2 ) 0 & ₆ (3+4 x-x^2 ) 1 aligned aligned & 3+4 x-x^2 6 & x^2-4 x+3 0 & (x-1)(x-3) 0 & x (1,3) & so a=1 & b=3 & ₀^2 [x^2 ] d x=? & I= ₀^1 [x^2 ] d x+ ₁^ 2 [x^2 ] d x+ _ 2 ^ 3 [x^2 ] d x+ _ 3 ^ 4 [x^2 ] d x & =0+|x|₁^ 2 +2|x|_ 2 ^ 3 +3|x|_ 3 ^ 4 aligned aligned & =( 2 -1)+2( 3 - 2 )+3(2- 3 ) & =5- 2 - 3 p + q + r =10 aligned