Question

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

Derivatives
ANSWERED
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, \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{-{x}}}^{{{x}}}}{\frac{{{\left.{d}{t}\right.}}}{{{t}^{{{10}}}+{1}}}}{i}{s}{\frac{{{1}}}{{{x}^{{{10}}}+{1}}}}\)
18
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours
...