COMEDK202510 May 2025Morning ShiftMathematicsIndefinite IntegrationActual
1 9+8 x-x^2 d x= (x)+c then (x)=
Options
- A1 10 | 4-x 4+x |
- B⁻¹ ( x-4 5 )
- C| 4-x 4+x |
- D1 5 ⁻¹ ( x-4 5 )
Correct answer
B. ⁻¹ ( x-4 5 )
Step-by-step solution
The integral is given by I = 1 9+8x-x^2 dx . Complete the square for the quadratic expression inside the square root: 9 + 8x - x^2 = 9 - (x^2 - 8x) = 9 - (x^2 - 8x + 16 - 16) = 9 - ((x-4)^2 - 16) = 9 + 16 - (x-4)^2 = 25 - (x-4)^2 . Substitute this into the integral: I = 1 25 - (x-4)^2 dx . Using the standard integral formula 1 a^2 - u^2 du = ⁻¹ ( u a ) + C , where a = 5 and u = x-4 , we get: I = ⁻¹ ( x-4 5 ) + C . Thus, (x) = ⁻¹ ( x-4 5 ) . Answer: ⁻¹ ( x-4 5 )