(11)
[tex]f(x) = e^{\sin^2 x} \\ \\ F(x) = \int_0^x f(t)\, dt \\ \\ \int\limits_{0}^{\pi/2}\cos 2x\cdot F(x) \, dx = \dfrac{1}{2}\int\limits_{0}^{\pi/2}(\sin2x)'\cdot F(x) \, dx = \\ \\ = \dfrac{\sin 2x\cdot F(x)}{2}\Big|_0^{\pi/2} - \dfrac{1}{2}\int\limits_{0}^{\pi/2} \sin 2x\cdot F'(x)\,dx = \\ \\ F'(x) = f(x) \\ \\= -\dfrac{1}{2}\int\limits_{0}^{\pi/2}\sin 2x\cdot f(x)\, dx=[/tex]
[tex]f'(x) = (\sin^2 x)'e^{\sin^2 x} = \sin 2x\cdot e^{\sin^2 x}= \sin 2x \cdot f(x)\\ \\\\ = -\dfrac{1}{2}\int\limits_{0}^{\pi/2}\sin 2x\cdot f(x) \, dx = \\ \\ = -\dfrac{1}{2}\int\limits_{0}^{\pi/2}f'(x)\, dx= \\ \\ = -\dfrac{f(x)}{2}\Big|_{0}^{\pi/2}= -\dfrac{e}{2}+\dfrac{1}{2}- = \boxed{\dfrac{1-e}{2}}[/tex]
