"Evaluate the following derivatives. ddx∫−xxdtt10+1" was answered by our community

Efan Halliday

Answered question

2021-02-22

Evaluate the following derivatives.

\frac{d}{d x} \int_{- x}^{x} \frac{d t}{t^{10} + 1}

Answer & Explanation

delilnaT

Skilled2021-02-24Added 94 answers

Step 1

g (x) = \int_{- x}^{x} \frac{d t}{t^{10} + 1}

Let
According to second Fundamental theorem of calculus, for a function,

F (x) = \int_{a}^{x} f (t) d t

which is continuous on [a, x].
F'(x)=f(x)
Step 2
So for function g(x) there exist an antiderivative such that

g^{'} (x) = \frac{d}{d x} \int_{- x}^{x} \frac{d t}{t^{10} + 1}

= \frac{1}{x^{10} + 1}

Thus, the derivative of the function,

\frac{d}{d x} \int_{- x}^{x} \frac{d t}{t^{10} + 1} i s \frac{1}{x^{10} + 1}

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