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  • The following functions are differentiable on (-1,2).Option: 1 <img alt="\mathrm{\int_x^{2 x}(\log t)^2 d t}" src="https://entrancecorner.oncodecogs.

The following functions are differentiable on (-1,2).

Option: 1

\mathrm{\int_x^{2 x}(\log t)^2 d t}


Option: 2

\mathrm{\int_0^{2 x} \frac{\sin t}{t} d t}


Option: 3

\mathrm{\int_0^x \frac{1-t+t^2}{1+t+t^2} d t}


Option: 4

none of these.


Answers (1)

best_answer

Since the functions  \mathrm{(\log t)^2} and   \mathrm{(\frac{\sin t}{t})}   are not defined on (-1,2), therefore the functions in (a) and (b) are not defined on (-1,2). The function \mathrm{ g(t)=\frac{1-t+t^2}{1+t+t^2}}  is continuous on (-1,2) and  \mathrm{ f(x)=\int_0^x \frac{1-t+t^2}{1+t+t^2} d t}  is the integral function of g(t). \mathrm{therefore~ f(x)} is differentiable on (-1,2) such that\mathrm{ f^{\prime}(x)=g(x)}.

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Irshad Anwar

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