MHT CET202521 Apr 2025Evening ShiftMathematicsIndefinite IntegrationActual
x 5 ^2 x+6 ^2 x ~d x=
Options
- A( x+ ^2 x+5 )+c , where c is the constant of integration
- B( x+ 6 ^2 x+5 )+c, where c is the constant of integration
- C- ( x+ ^2 x+6 )+c, where c is the constant of integration
- D- ( x+ ^2 x+5 )+c , where c is the constant of integration
Correct answer
D. - ( x+ ^2 x+5 )+c , where c is the constant of integration
Step-by-step solution
The integral I = x 5 ^2 x+6 ^2 x dx simplifies by expressing the denominator in terms of ^2 x : 5 ^2 x + 6 ^2 x = 5( ^2 x + ^2 x) + ^2 x = 5 + ^2 x Substituting t = x with dt = - x dx transforms the integral: I = - dt t^2 + 5 Using the standard integral formula dx x^2 + a^2 = |x + x^2 + a^2 | + C : I = - |t + t^2 + 5 | + C Reversing the substitution yields: I = - | x + ^2 x + 5 | + C The expression inside the logarithm remains positive since ^2 x + 5 > | x| for all real x . This result corresponds to option D.