AP EAMCET201823 Apr 2018Morning ShiftMathematicsIndefinite IntegrationActual
x-1 (x+1) x (x^2+x+1 ) d x=
Options
- A⁻¹ ( x^2+x+1 x )+c
- B2 ⁻¹ ( x^2+x+1 x )+c
- C⁻¹ ( x^2+x+1 x )+c
- D2 ⁻¹ ( x+ 1 x +1 )+c
Correct answer
D. 2 ⁻¹ ( x+ 1 x +1 )+c
Step-by-step solution
aligned & Let, I= x-1 (x+1) x (x^2+x+1 ) d x & = (x-1)(x+1) (x+1)^2 x (x^2+x+1 ) d x & = x^2-1 (x+1)^2 x 1+x+ 1 x d x & = x^2-1 x^2 ( x+1 x )^2 x 1+x+ 1 x d x aligned aligned & = (1- 1 x^2 ) (1+ 1 x^2 + 2 x ) x 1+x+ 1 x d x & = 1- 1 x^2 (2+ 1 x +x ) 1+x+ 1 x d x & = (1- 1 x^2 ) 1+ (x+ 1 x +1 ) 1+x+ 1 x d x & = 1- 1 x^2 1+ (x+ 1 x +1 )^2 1+x+ 1 x d x & Let x+ 1 x +1 =t & (1- 1 x^2 ) 2 x+ 1 x +1 d x=d t & 1= 2 d t 1+t^2 & =2 ⁻¹ t+c=2 ⁻¹ ( x+ 1 x +1 )+c & aligned