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# Evaluate the following derivatives. \frac{d}{dx}\int_{-x}^{x}\frac{dt}{t^{10}+1}

Derivatives
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asked 2021-02-22
Evaluate the following derivatives.
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{-{x}}}^{{{x}}}}{\frac{{{\left.{d}{t}\right.}}}{{{t}^{{{10}}}+{1}}}}$$

## Expert Answers (1)

2021-02-24
Step 1
$$\displaystyle{g{{\left({x}\right)}}}={\int_{{-{x}}}^{{{x}}}}{\frac{{{\left.{d}{t}\right.}}}{{{t}^{{{10}}}+{1}}}}$$
Let
According to second Fundamental theorem of calculus, for a function,
$$\displaystyle{F}{\left({x}\right)}={\int_{{{a}}}^{{{x}}}}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}$$
which is continuous on [a, x].
F'(x)=f(x)
Step 2
So for function g(x) there exist an antiderivative such that
$$\displaystyle{g}'{\left({x}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{-{x}}}^{{{x}}}}{\frac{{{\left.{d}{t}\right.}}}{{{t}^{{{10}}}+{1}}}}$$
$$\displaystyle={\frac{{{1}}}{{{x}^{{{10}}}+{1}}}}$$
Thus, the derivative of the function, $$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{-{x}}}^{{{x}}}}{\frac{{{\left.{d}{t}\right.}}}{{{t}^{{{10}}}+{1}}}}{i}{s}{\frac{{{1}}}{{{x}^{{{10}}}+{1}}}}$$

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