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Question Number 202882 by dimentri last updated on 05/Jan/24

$$\downharpoonleft \underset{―}{}$$

Answered by cortano12 last updated on 05/Jan/24

$$\{\begin{array}{l}5\underset{3}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=10\\ 5\underset{1}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=2\end{array}$$$$\Rightarrow \underset{1}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\underset{1}{\stackrel{3}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+\underset{3}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$$$$\Rightarrow 5\underset{1}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=5\underset{1}{\stackrel{3}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+5\underset{3}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$$$$\Rightarrow 2=5\underset{1}{\stackrel{3}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+10$$$$\Rightarrow 5\underset{1}{\stackrel{3}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=-8$$

Answered by MM42 last updated on 05/Jan/24

$${\int }_1^{3}5f={\int }_1^{6}5f+{\int }_6^{3}5f=2-10=-8✓$$

Answered by aba last updated on 05/Jan/24

$${\int }_1^6\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_1^3\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_3^6\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$$$$\Rightarrow {\int }_1^3\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_1^6\mathrm{f}\left(\mathrm{x}\right)-{\int }_3^6\mathrm{f}\left(3\right)\mathrm{dx}$$$$\Rightarrow 5{\int }_1^3\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=5.\frac{2}{5}-5.2$$$$\Rightarrow {\int }_1^{3}5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=-8✓✓$$

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