Answer
$\dfrac{99}{10}-2 \ln 10$
Work Step by Step
As we are given that $\int^{\ln 10}_{0} 4\sinh^2(\dfrac{x}{2}) dx$
Use formula, $\sinh x=\dfrac{e^{x} - e^{-x}}{2}$
Thus, $\int^{\ln 10}_{0} 4\sinh^2(\dfrac{x}{2}) dx=\int^{\ln 10}_{0} 4 [\dfrac{e^{x/2} - e^{-x/2}}{2}]^2 dx$
and
$\int^{\ln 10}_{0} 4 [\dfrac{e^{x/2} - e^{-x/2}}{2}]^2 dx= \int^{\ln 10}_{0} (e^{x} + e^{-x} -2) dx=[ e^{x} - e^{-x}-2x]^{\ln 10}_{0}$
or, $=(10 -\dfrac{1}{10}-2 \ln 10)$
Hence, $\int^{\ln 10}_{0} 4\sinh^2(\dfrac{x}{2}) dx=\dfrac{99}{10}-2 \ln 10$
