MHT CET202522 Apr 2025Morning ShiftMathematicsIndefinite IntegrationActual
2 x 2 x 4- ^4 2 x ~d x=
Options
- A1 4 ⁻¹ ( ^2 2 x 2 )+c , where c is the constant of integration.
- B-1 4 ⁻¹ ( ^2 2 x 2 )+c , where c is the constant of integration.
- C1 2 ⁻¹ ( ^2 2 x 2 )+c , where c is the constant of integration.
- D-1 2 ⁻¹ ( ^2 2 x 2 )+c , where c is the constant of integration.
Correct answer
B. -1 4 ⁻¹ ( ^2 2 x 2 )+c , where c is the constant of integration.
Step-by-step solution
Let I = 2x 2x 4 - ^4 2x dx . Applying the substitution u = ^2 2x , the differential becomes du = -4 2x 2x dx , yielding 2x 2x dx = - 1 4 du . Substituting into the integral gives I = - 1 4 1 4 - u^2 du . This evaluates to the inverse sine form: I = - 1 4 ⁻¹ ( u 2 ) + c . Returning to the original variable, I = - 1 4 ⁻¹ ( ^2 2x 2 ) + c . Final Answer: B