MHT CET20239 May 2023Morning ShiftMathematicsIndefinite IntegrationActual
e ^ ⁻¹ x 1+x^2 [ ( ⁻¹ 1+x^2 )^2+ ⁻¹ ( 1-x^2 1+x^2 ) ] d x, x>0=
Options
- A( ⁻¹ x )^2 e ^ ⁻¹ x + c , where c is a constant of integration.
- B( ⁻¹ x ) e ^ ⁻¹ x + c , where c is a constant of integration.
- C( ⁻¹ x ) e ^ 2 ⁻¹ x + c , where c is a constant of integration.
- D( ⁻¹ x )^2 e ^ 2 ⁻¹ x + c , where c is a constant of integration.
Correct answer
A. ( ⁻¹ x )^2 e ^ ⁻¹ x + c , where c is a constant of integration.
Step-by-step solution
aligned & e ^ ⁻¹ x 1+x^2 [ ( ⁻¹ 1+x^2 )^2+ ⁻¹ ( 1-x^2 1+x^2 ) ] d x, & Put x= t & d x= ^2 t d t aligned aligned I & = e ^ ⁻¹( t ) 1+ ^2 t [ ( ⁻¹ 1+ ^2 t )^2+ ⁻¹ ( 1- ^2 t 1+ ^2 t ) ] & = e ^ t ^2 t [ ( ^2 t ( t ) )^2+ ⁻¹( 2 t ) ] ^2 t d t & = e ^ t [ t ^2+2 t ] dt & = e ^ t t ^2+ c [ e ^x f (x) f ^ (x)= e ^x f (x)+ c ] & = t ^2 e ^ t + c & = ( ⁻¹ x )^2 e ^ tan ⁻¹ x + c . aligned