MHT CET202416 May 2024Evening ShiftMathematicsIndefinite IntegrationActual
e ^ ⁻¹ x 1+x^2 [ ( ⁻¹ 1+x^2 )^2+ ⁻¹ ( 1-x^2 1+x^2 ) ] d x , where x 0 is
Options
- A( ⁻¹ x ) e ^ ⁻¹ x + c , where c is a constant of integration.
- B( ⁻¹ x )^2 e ^ ⁻¹ x + c , where c is a constant of integration.
- C2 ( ⁻¹ x ) e ^ ⁻¹ x + c , where c is a constant of integration.
- D2 ( ⁻¹ x )^2 e ^ ⁻¹ x + c , where c is a constant of integration.
Correct answer
B. ( ⁻¹ x )^2 e ^ ⁻¹ x + c , where c is a constant of integration.
Step-by-step solution
Let I = e ^ ⁻¹ x 1+x^2 [ ( ⁻¹ 1+x^2 )^2+ ⁻¹ ( 1-x^2 1+x^2 ) ] d x array ll & Put x= t & ~d x= ^2 t d dt array array r I= e^ ⁻¹( t) 1+ ^2 t [ ( ⁻¹ 1+ ^2 t )^2+ ⁻¹ ( 1- ^2 t 1+ ^2 t ) ] ^2 t d t array aligned & = e^t ^2 t [ ( ⁻¹( t) )^2+ ⁻¹( 2 t) ] ^2 t d t & = e^t [t^2+2 t ] d t & =e^t t^2+c aligned aligned & = t ^2 e ^ t + c ^ [ [ e ^x [ f (x)+ f ^ (x) ] d x= e ^x f (x)+ c ] . & = ( ⁻¹ x )^2 e ^ ⁻¹ x + c aligned