вправа 7.1 гдз 11 клас математика Нелін Долгова 2019
Вправа 7.1
Умова:
Умова:
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Відповідь ГДЗ:
\begin{equation} 1)\int_{-1}^{2}x^{4}dx=\frac{x^{5}}{5} |_{2}^{-1}= \end{equation} \begin{equation} =\frac{2^{5}}{5}-\frac{(-1)^{5}}{5}= \end{equation} \begin{equation} =\frac{32}{5}+\frac{1}{5}=\frac{33}{5}=6,6; \end{equation} \begin{equation} 2)\int_{0}^{\frac{\Pi }{2}}cosxdx=sin|_{0}^{\frac{\Pi }{2}}= \end{equation} \begin{equation} =sin\frac{\Pi }{2}-sin0=1; \end{equation} \begin{equation} 3)\int_{1}^{3}x^{3}dx=\frac{x^{4}}{4}|_{1}^{3}= \end{equation} \begin{equation} =\frac{3^{4}}{4}-\frac{1}{4}=\frac{81}{4}-\frac{1}{4}= \end{equation} \begin{equation} =\frac{80}{4}=20; \end{equation} \begin{equation} 4)\int_{\frac{\Pi }{4}}^{\frac{\Pi }{2}}\frac{dx}{sin^{2}x}= \end{equation} \begin{equation} =-ctgx|_{{\frac{\Pi }{4}}}^{{\frac{\Pi }{4}}}= \end{equation} \begin{equation} =-ctg\frac{\Pi }{2}+ctg\frac{\Pi }{4}=1; \end{equation} \begin{equation} 5)\int_{1}^{2}\frac{dx}{(2x+1)^{2}}= \end{equation} \begin{equation} =-\frac{1}{2(2x+1)}|_{1}^{2}= \end{equation} \begin{equation} =-\frac{1}{1(2\cdot 2+1)}+\frac{1}{2(2+1)}= \end{equation} \begin{equation} =-\frac{1}{10}+\frac{1}{6}=\frac{-3+5}{30}= \end{equation} \begin{equation} =\frac{2}{30}=\frac{1}{15}; \end{equation} \begin{equation} 6)\int_{0}^{\Pi }3cos\frac{x}{2}dx= \end{equation} \begin{equation} =3\cdot 2sin\frac{x}{2}\int_{0}^{\Pi }= \end{equation} \begin{equation} =6(sin\frac{\Pi }{2}-sin0)=6; \end{equation} \begin{equation} 7)\int_{1}^{10}\frac{dx}{x^{2}}=-\frac{1}{x}|_{1}^{10}= \end{equation} \begin{equation} =-\frac{1}{10}+1=0,9; \end{equation} \begin{equation} 8)\int_{\frac{\Pi }{4}}^{\frac{\Pi }{2}}sin2xdx= \end{equation} \begin{equation} =-\frac{1}{2}cos2x|_{\frac{\Pi }{4}}^{\frac{\Pi }{2}}= \end{equation} \begin{equation} =-\frac{1}{2}cos\frac{\Pi }{2}\cdot 2+\frac{1}{2}cos2\cdot \frac{\Pi }{4}= \end{equation} \begin{equation} =-\frac{1}{2}cos\Pi +\frac{1}{2}cos\frac{\Pi }{2}=\frac{1}{2}. \end{equation} Відповідь: 1) 6,6; 2) 1; 3) 20; 4) 1; \begin{equation} 5)\frac{1}{15}; \end{equation} 6) 6; 7) 0,9; 8) 0,5.
\begin{equation} 1)\int_{-1}^{2}x^{4}dx=\frac{x^{5}}{5} |_{2}^{-1}= \end{equation} \begin{equation} =\frac{2^{5}}{5}-\frac{(-1)^{5}}{5}= \end{equation} \begin{equation} =\frac{32}{5}+\frac{1}{5}=\frac{33}{5}=6,6; \end{equation} \begin{equation} 2)\int_{0}^{\frac{\Pi }{2}}cosxdx=sin|_{0}^{\frac{\Pi }{2}}= \end{equation} \begin{equation} =sin\frac{\Pi }{2}-sin0=1; \end{equation} \begin{equation} 3)\int_{1}^{3}x^{3}dx=\frac{x^{4}}{4}|_{1}^{3}= \end{equation} \begin{equation} =\frac{3^{4}}{4}-\frac{1}{4}=\frac{81}{4}-\frac{1}{4}= \end{equation} \begin{equation} =\frac{80}{4}=20; \end{equation} \begin{equation} 4)\int_{\frac{\Pi }{4}}^{\frac{\Pi }{2}}\frac{dx}{sin^{2}x}= \end{equation} \begin{equation} =-ctgx|_{{\frac{\Pi }{4}}}^{{\frac{\Pi }{4}}}= \end{equation} \begin{equation} =-ctg\frac{\Pi }{2}+ctg\frac{\Pi }{4}=1; \end{equation} \begin{equation} 5)\int_{1}^{2}\frac{dx}{(2x+1)^{2}}= \end{equation} \begin{equation} =-\frac{1}{2(2x+1)}|_{1}^{2}= \end{equation} \begin{equation} =-\frac{1}{1(2\cdot 2+1)}+\frac{1}{2(2+1)}= \end{equation} \begin{equation} =-\frac{1}{10}+\frac{1}{6}=\frac{-3+5}{30}= \end{equation} \begin{equation} =\frac{2}{30}=\frac{1}{15}; \end{equation} \begin{equation} 6)\int_{0}^{\Pi }3cos\frac{x}{2}dx= \end{equation} \begin{equation} =3\cdot 2sin\frac{x}{2}\int_{0}^{\Pi }= \end{equation} \begin{equation} =6(sin\frac{\Pi }{2}-sin0)=6; \end{equation} \begin{equation} 7)\int_{1}^{10}\frac{dx}{x^{2}}=-\frac{1}{x}|_{1}^{10}= \end{equation} \begin{equation} =-\frac{1}{10}+1=0,9; \end{equation} \begin{equation} 8)\int_{\frac{\Pi }{4}}^{\frac{\Pi }{2}}sin2xdx= \end{equation} \begin{equation} =-\frac{1}{2}cos2x|_{\frac{\Pi }{4}}^{\frac{\Pi }{2}}= \end{equation} \begin{equation} =-\frac{1}{2}cos\frac{\Pi }{2}\cdot 2+\frac{1}{2}cos2\cdot \frac{\Pi }{4}= \end{equation} \begin{equation} =-\frac{1}{2}cos\Pi +\frac{1}{2}cos\frac{\Pi }{2}=\frac{1}{2}. \end{equation} Відповідь: 1) 6,6; 2) 1; 3) 20; 4) 1; \begin{equation} 5)\frac{1}{15}; \end{equation} 6) 6; 7) 0,9; 8) 0,5.