📝 题目
5.5.4 求 ${F}^{\prime }\left( x\right)$ :
(1) $F\left( x\right) = {\int }_{\sin x}^{\cos x}{\mathrm{e}}^{x\sqrt{1 - {y}^{2}}}\mathrm{\;d}y$ ; (2) $F\left( x\right) = {\int }_{a + x}^{b + x}\frac{\sin {xy}}{y}\mathrm{\;d}y$ ;
(3) $F\left( x\right) = {\int }_{0}^{x}{\int }_{{t}^{2}}^{{x}^{2}}f\left( {t,s}\right) \mathrm{d}s\mathrm{\;d}t$ .
💡 答案与解析
5.5.4 (1) $\displaystyle{- \sin x{\mathrm{e}}^{x\left| {\sin x}\right| } - \cos x{\mathrm{e}}^{x\left| {\cos x}\right| } + {\int }_{\sin x}^{\cos x}\sqrt{1 - {y}^{2}}{\mathrm{e}}^{x\sqrt{1 - {y}^{2}}}\mathrm{\;d}y}$ ;
(2) $\left( {\frac{1}{x} + \frac{1}{b + x}}\right) \sin x\left( {b + x}\right) - \left( {\frac{1}{x} + \frac{1}{a + x}}\right) \sin x\left( {a + x}\right)$ ；
(3) ${2x}{\int }_{0}^{x}f\left( {t,{x}^{2}}\right) \mathrm{d}t$ .