MHT CET202616 April 2026Evening ShiftMathematicsIndefinite IntegrationActual
x^4 + 1 x^6 + 1 dx =
Options
- A⁻¹x - 1 3 ⁻¹(x^3) + c
- B⁻¹x + 1 3 ⁻¹(x^3) + c
- C⁻¹x - ⁻¹(x^3) + c
- D⁻¹x + ⁻¹(x^3) + c
Correct answer
B. ⁻¹x + 1 3 ⁻¹(x^3) + c
Step-by-step solution
The given integral is I = x^4 + 1 x^6 + 1 dx The denominator can be factored as x^6 + 1 = (x^2)^3 + 1 = (x^2 + 1)(x^4 - x^2 + 1) Rewriting the numerator as x^4 + 1 = (x^4 - x^2 + 1) + x^2 I = (x^4 - x^2 + 1) + x^2 (x^2 + 1)(x^4 - x^2 + 1) dx Separating the terms: I = x^4 - x^2 + 1 (x^2 + 1)(x^4 - x^2 + 1) dx + x^2 x^6 + 1 dx I = 1 x^2 + 1 dx + 1 3 3x^2 (x^3)^2 + 1 dx Substituting t = x^3 dt = 3x^2 dx in the second integral: I = ⁻¹x + 1 3 1 t^2 + 1 dt I = ⁻¹x + 1 3 ⁻¹(t) + c I = ⁻¹x + 1 3 ⁻¹(x^3) + c Answer: ⁻¹x + 1