AP EAMCET20225 Jul 2022Morning ShiftMathematicsIndefinite IntegrationActual
e^ ⁻¹ x 1+x^2 [ ( ⁻¹ 1+x^2 )^2+ ⁻¹ ( 1-x^2 1+x^2 ) ] d x=
Options
- Ae^ ⁻¹ x ( ⁻¹ x )^2+C
- Be^ ⁻¹ x ( ⁻¹ x )^2+C
- Ce^ ⁻¹ x ( ⁻¹ ( 1+x^2 ) )+C
- De^ ⁻¹ x ( ⁻¹ ( 1-x^2 1+x^2 ) )+C
Correct answer
A. e^ ⁻¹ x ( ⁻¹ x )^2+C
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 . On putting ⁻¹ x=t and d x 1+x^2 =d t , we get I= e^t [t^2+2 t ] d t I= (t^2 e^t+2 e^t t ) d t By integration by parts, we get aligned & I=t^2 e^t- 2 t e^t d t- 2 t e^t d t+C & I=t^2 e^t+C=e^ ⁻¹(x) ( ⁻¹ x )^2+C aligned