MHT CET202618 April 2026Evening ShiftMathematicsIndefinite IntegrationActual
If e^ x + ⁻¹x ( x^2 + 2 ^2( ⁻¹x) )dx = e^ f(x) + c , then
Options
- Af(x) is strictly decreasing on R .
- Bf(x) is strictly increasing on R^+ and strictly decreasing on R^- .
- Cf(x) is strictly increasing on R .
- Df(x) is strictly decreasing on R^+ and strictly increasing on R^- .
Correct answer
C. f(x) is strictly increasing on R .
Step-by-step solution
The given integral is I = e^ x + ⁻¹x ( x^2 + 2 ^2( ⁻¹x) )dx . Using the identity ^2 = 1 + ^2 , we have ^2( ⁻¹x) = 1 + ^2( ⁻¹x) = 1 + x^2 . Substituting this into the integral gives: I = e^ x + ⁻¹x ( x^2 + 2 1 + x^2 )dx I = e^ x + ⁻¹x ( x^2 + 1 + 1 1 + x^2 )dx I = e^ x + ⁻¹x (1 + 1 1 + x^2 )dx Let t = x + ⁻¹x . Differentiating with respect to x , we get: dt = (1 + 1 1 + x^2 )dx The integral becomes: I = e^t dt = e^t + c Substituting back t = x + ⁻¹x , we get: I = e^ x + ⁻¹x + c Comparing this with the given expressi