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

Evaluate the following derivatives.
$\frac{d}{dx}{\int }_{-x}^{x}\frac{dt}{{t}^{10}+1}$
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Step 1
$g\left(x\right)={\int }_{-x}^{x}\frac{dt}{{t}^{10}+1}$
Let
According to second Fundamental theorem of calculus, for a function,
$F\left(x\right)={\int }_{a}^{x}f\left(t\right)dt$
which is continuous on [a, x].
F'(x)=f(x)
Step 2
So for function g(x) there exist an antiderivative such that
${g}^{\prime }\left(x\right)=\frac{d}{dx}{\int }_{-x}^{x}\frac{dt}{{t}^{10}+1}$
$=\frac{1}{{x}^{10}+1}$
Thus, the derivative of the function, $\frac{d}{dx}{\int }_{-x}^{x}\frac{dt}{{t}^{10}+1}is\frac{1}{{x}^{10}+1}$