AP EAMCET202124 Aug 2021Morning ShiftMathematicsDefinite IntegrationActual
If b>a , then _a^b d x (x-a)(b-x) is equal to
Options
- A/ 2
- B/ 3
- C/ 6
Correct answer
3
Step-by-step solution
_a^b d x (x-a)(b-x) aligned & _a^b d x -x^2+(a+b) x-a b & = _a^b d x - (x^2-(a+b) x+a b . & = _a^b d x - [x^2+ ( a+b 2 )^2+a b- ( a+b 2 )^2 ]-(a+b) x aligned aligned & = _a^b d x ( a-b 2 )^2- x- ( a+b 2 ) ^2 [ b>a] & = _a^b d x ( b-a 2 )^2- x- ( a+b 2 ) ^2 & d x a^2-x^2 = ⁻¹ x a +C aligned aligned & = ⁻¹ [ x- ( a+b 2 ) b-a 2 ] ]_a^b & = ⁻¹ [ b- ( a+b 2 ) b-a 2 ]- ⁻¹ ( a- a+b 2 b-a 2 ) & = ⁻¹(+1)- ⁻¹(-1)=+ 2 + 2 =+ aligned