If $\large \int e^{ax} cos bx dx = \frac{e^{2x}}{2a}f(x) + c $ , then

Q: If $\large \int e^{ax} cos bx dx = \frac{e^{2x}}{2a}f(x) + c $ , then

(A) $\large \frac{f”(x)}{f(x)} = 24 $

(B) $\large \frac{f”(x)}{f(x)} = 25 $

(C) range of f(x) is $ (-\sqrt{29} , \sqrt{29}) $

(D) $\large \frac{f(x)}{f'(x)} = e^{cosbx} (a sinx)$

Sol: $\large \int e^{ax} cos bx dx = \frac{e^{ax}}{a^2 + b^2}(a cosbx + b sinbx) + c $

$\large \frac{e^{2x}}{2^2 + 5^2}f(x) + c = \frac{e^{ax}}{a^2 + b^2}(a cosbx + b sinbx) + c $

$\large \frac{e^{2x}}{29}f(x) + c = \frac{e^{ax}}{a^2 + b^2}(a cosbx + b sinbx) + c $

a = 2 , b = 5 and f(x) = 2 cos5x + 5 sin5x

Range of f(x) is $ (-\sqrt{29} , \sqrt{29}) $

Hence (C) is the correct answer.