\begin{align} E&=\int_0^xFdx =\int_0^x\frac{d}{dt}(mv)dx =\int_0^t\frac{d}{dt}(mv)vdt =\int_0^{mv}vd(mv) =\int_0^v vd\left(\frac{m_0v}{\sqrt{1-(v/c)^2}}\right)\\ &=m_0\int_0^v\left(\frac{v}{[1-(v/c)^2]^{1/2}}+\frac{v^3/c^2}{[1-(v/c)^2]^{3/2}}\right)dv =m_0\int_0^v\frac{vdv}{[1-(v/c)^2]^{3/2}}\\ &=m_0c^2\left(\frac{1}{[1-(v/c)^2]^{1/2}}-1\right) =(mc^2-m_0c^2) =(m-m_0)c^2\Rightarrow \end{align}