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Question Number 92608 by jagoll last updated on 08/May/20

Commented by john santu last updated on 08/May/20

g(t) = { ((−(1/2)t ,−4≤t≤0)),((−(√(9−(t−3)^2 )), 0≤t≤6)),((3 , 6<t≤8)) :} f(x)=3x+∫_0 ^x g(t) dt f(x)=3x+ { ((−(1/4)x^2 )),((−(9/2)sin^(−1) (((x−3)/3))−(1/2)(((x−3)/3))(√(6x−x^2 ))+((9π)/4))),((6x )) :}

$$\mathrm{g}\left(\mathrm{t}\right)\:=\:\begin{cases}{−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{t}\:,−\mathrm{4}\leqslant\mathrm{t}\leqslant\mathrm{0}}\\{−\sqrt{\mathrm{9}−\left(\mathrm{t}−\mathrm{3}\right)^{\mathrm{2}} },\:\mathrm{0}\leqslant\mathrm{t}\leqslant\mathrm{6}}\\{\mathrm{3}\:,\:\mathrm{6}<\mathrm{t}\leqslant\mathrm{8}}\end{cases} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{3x}+\underset{\mathrm{0}} {\overset{\mathrm{x}} {\int}}\:\mathrm{g}\left(\mathrm{t}\right)\:\mathrm{dt}\: \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{3x}+\begin{cases}{−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{x}^{\mathrm{2}} }\\{−\frac{\mathrm{9}}{\mathrm{2}}\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{x}−\mathrm{3}}{\mathrm{3}}\right)−\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{x}−\mathrm{3}}{\mathrm{3}}\right)\sqrt{\mathrm{6x}−\mathrm{x}^{\mathrm{2}} }+\frac{\mathrm{9}\pi}{\mathrm{4}}}\\{\mathrm{6x}\:}\end{cases} \\ $$

Commented by jagoll last updated on 08/May/20

find f(x)= 3x+∫_0 ^x g(t) dt graph g(x) it shown above

$$\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{3x}+\underset{\mathrm{0}} {\overset{\mathrm{x}} {\int}}\:\mathrm{g}\left(\mathrm{t}\right)\:\mathrm{dt} \\ $$$$\mathrm{graph}\:\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{it}\:\mathrm{shown}\:\mathrm{above} \\ $$

Commented by MJS last updated on 08/May/20

∫_t ...dt is not possible

$$\underset{{t}} {\int}...{dt}\:\mathrm{is}\:\mathrm{not}\:\mathrm{possible} \\ $$

Commented by jagoll last updated on 08/May/20

why sir? f(x) = 3x+∫_0 ^( x) g(t) dt sir

$$\mathrm{why}\:\mathrm{sir}? \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{3x}+\int_{\mathrm{0}} ^{\:\mathrm{x}} \:\mathrm{g}\left(\mathrm{t}\right)\:\mathrm{dt}\:\mathrm{sir} \\ $$

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