MHT CET202525 Apr 2025Evening ShiftMathematicsIndefinite IntegrationActual
x^4 ( ⁻¹ x^5 ) 1+x¹⁰ ~d x equals
Options
- A( ⁻¹ x^5 ) 5 +c , where c is the constant of integration
- Bx^4 ( ⁻¹ x^5 )+ c , where c is the constant of integration
- C( ⁻¹ x^5 ) 4 +c , where c is the constant of integration
- D( ⁻¹ x^5 )+ c , where c is the constant of integration
Correct answer
A. ( ⁻¹ x^5 ) 5 +c , where c is the constant of integration
Step-by-step solution
Evaluate I = x^4 ( ⁻¹ x^5) 1+x¹⁰ d x using substitution. Let u = ⁻¹ x^5 , so u = x^5 . Differentiate: du dx = 1 1+(x^5)^2 5x^4 = 5x^4 1+x¹⁰ . Rewriting gives x^4 1+x¹⁰ dx = 1 5 du . Substitute into I : I = u 1 5 du = 1 5 u + C . Back-substitute u = ⁻¹ x^5 to obtain I = 1 5 ( ⁻¹ x^5) + C . The result corresponds with option A.