MHT CET202620 April 2026Morning ShiftMathematicsIndefinite IntegrationActual
x^4(x¹⁰ - 1) x²⁰ + 3x¹⁰ + 1 , dx =
Options
- A⁻¹ (x^5 + 1 x^5 ) + c
- B1 5 ⁻¹ (x^5 + 1 x^5 ) + c
- C⁻¹ (x¹⁰ + 1 x¹⁰ ) + c
- D1 10 ⁻¹ (x¹⁰ + 1 x¹⁰ ) + c
Correct answer
B. 1 5 ⁻¹ (x^5 + 1 x^5 ) + c
Step-by-step solution
I = x^4(x¹⁰ - 1) x²⁰ + 3x¹⁰ + 1 , dx Dividing the numerator and the denominator by x¹⁰ : I = x^4 - 1 x^6 x¹⁰ + 3 + 1 x¹⁰ , dx Let x^5 + 1 x^5 = t Differentiating with respect to x : (5x^4 - 5 x^6 ) dx = dt (x^4 - 1 x^6 ) dx = dt 5 The denominator can be written as: x¹⁰ + 1 x¹⁰ + 3 = (x^5 + 1 x^5 )^2 - 2 + 3 = t^2 + 1 Substituting these into the integral: I = 1 t^2 + 1 dt 5 I = 1 5 ⁻¹(t) + c I = 1 5 ⁻¹ (x^5 + 1 x^5 ) + c Answer: 1 5 ⁻¹ (x^5 + 1 x^5 ) + c