AP EAMCET201922 Apr 2019Morning ShiftMathematicsIndefinite IntegrationActual
( x x ( x^2-1 )^3 d x= )
Options
- A( ⁻¹ x+ x x^2-1 +C )
- B( ⁻¹ x- x x^2-1 +C )
- C( x x^2-1 - ⁻¹ x+C )
- D( - x x^2-1 - ⁻¹ x+C )
Correct answer
B. ( ⁻¹ x- x x^2-1 +C )
Step-by-step solution
Let (I= x x ( x^2-1 )^3 d x ) Let ( x^2-1 =t 2 x d x 2 x^2-1 =d t x d x x^2-1 =d t ) ( aligned & I= x d t t^2 &= ( 1+t^2 ) t^2 d t= (1+t^2 ) ( 1 t^2 ) d t & . [ x^2-1 =t ] x^2=t^2+1 x= 1+t^2 ] &=- 1 t 1+t^2 - 2 t 2 1+t^2 (- 1 t ) 1 1+t^2 d t &=- 1 t 1+t^2 + d t 1+t^2 &=- 1 t 1+t^2 + ⁻¹(t)+c & ⁻¹ ( x^2-1 )- x x^2-1 +c &= ⁻¹(x)- x x^2-1 +c aligned )