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Sagot :
The limits in the integral of the options are missing. They are :
a). [tex]\frac{d}{dx} \int^2_0 f(x)dx = 0[/tex]
b). [tex]\frac{d}{dx} \int^x_2 f(t)dt = 0[/tex]
c). [tex]$\int^x_2f'(x)dx=f(x)$[/tex]
Solution:
We known that
[tex]$\frac{d}{dx}\int^{g(x)}_{h(x)}f(t)dt = f(g(x))\cdot g'(x)-f(h(x))\cdot h'(x)$[/tex]
a). [tex]\frac{d}{dx} \int^2_0 f(x)dx[/tex]
[tex]$=f(2) \cdot \frac{d}{dx}(2) - f(0)\cdot \frac{d}{dx}(0)$[/tex]
= 0 - 0
= 0
Hence it is true.
b). [tex]\frac{d}{dx} \int^x_2 f(t)dt[/tex]
[tex]$=f(x)\frac{d}{dx}(x)-f(2)\cdot \frac{d}{dx}(2)$[/tex]
[tex]$=f(x) \cdot 1 -f(2)\times 0 $[/tex]
= f(x)
Hence it is true.
c). [tex]$\int^x_2f'(x)dx[/tex]
[tex]$\left[f(x)\right]^x_2 = f(x)-f(2)$[/tex]
[tex]$\neq f(x) $[/tex]
Hence it is false.
Therefore option a) and b). are true.
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