Question Number 70571 by ahmadshahhimat775@gmail.com last updated on 05/Oct/19
Commented by mathmax by abdo last updated on 05/Oct/19
![∃ c ∈]3,3+h[ /∫_3 ^(h+3) ((5dx)/(x^3 +7)) =(5/(c^3 +7)) ∫_3 ^(h+3) dx=((5h)/(c^3 +7)) ⇒ (1/h) ∫_3 ^(h+3) ((5dx)/(x^3 +7)) =(5/(c^3 +7)) h→0 ⇒c→3 ⇒ lim_(h→0) (1/h) ∫_3 ^(h+3) ((5dx)/(x^3 +7)) =(5/(3+7)) =(5/(10)) =(1/2)](Q70573.png)
$$\left.\exists\:{c}\:\in\right]\mathrm{3},\mathrm{3}+{h}\left[\:/\int_{\mathrm{3}} ^{{h}+\mathrm{3}} \:\frac{\mathrm{5}{dx}}{{x}^{\mathrm{3}} \:+\mathrm{7}}\:=\frac{\mathrm{5}}{{c}^{\mathrm{3}} \:+\mathrm{7}}\:\int_{\mathrm{3}} ^{{h}+\mathrm{3}} {dx}=\frac{\mathrm{5}{h}}{{c}^{\mathrm{3}} \:+\mathrm{7}}\:\Rightarrow\right. \\ $$$$\frac{\mathrm{1}}{{h}}\:\int_{\mathrm{3}} ^{{h}+\mathrm{3}} \:\frac{\mathrm{5}{dx}}{{x}^{\mathrm{3}} \:+\mathrm{7}}\:=\frac{\mathrm{5}}{{c}^{\mathrm{3}} \:+\mathrm{7}}\:\:\:\:\:\:{h}\rightarrow\mathrm{0}\:\Rightarrow{c}\rightarrow\mathrm{3}\:\Rightarrow \\ $$$${lim}_{{h}\rightarrow\mathrm{0}} \:\:\:\frac{\mathrm{1}}{{h}}\:\int_{\mathrm{3}} ^{{h}+\mathrm{3}} \:\frac{\mathrm{5}{dx}}{{x}^{\mathrm{3}} \:+\mathrm{7}}\:=\frac{\mathrm{5}}{\mathrm{3}+\mathrm{7}}\:=\frac{\mathrm{5}}{\mathrm{10}}\:=\frac{\mathrm{1}}{\mathrm{2}} \\ $$