Evaluate the following derivatives.

$\frac{d}{dx}{\int}_{-x}^{x}\frac{dt}{{t}^{10}+1}$

Efan Halliday
2021-02-22
Answered

Evaluate the following derivatives.

$\frac{d}{dx}{\int}_{-x}^{x}\frac{dt}{{t}^{10}+1}$

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delilnaT

Answered 2021-02-24
Author has **94** answers

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,$\frac{d}{dx}{\int}_{-x}^{x}\frac{dt}{{t}^{10}+1}is\frac{1}{{x}^{10}+1}$

Let

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

which is continuous on [a, x].

F'(x)=f(x)

Step 2

So for function g(x) there exist an antiderivative such that

Thus, the derivative of the function,

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(a) f ' (0)1

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(a) f ' (0)1

(b) f ' (1)2

(c) f ' (2)3

(d) f ' (3)4

(e) f ' (4)5

(f) f ' (5)6

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