To determine

To find:

Derivative of thegiven function

Answer

F'x=-x+sin⁡x

Explanation

1) Concept:

i) First Fundamental Theorem of Calculus

 If f is continuous on a, b, then function g defined by

gx=∫axf(t)dt

is continuous on a, b and differentiable on (a, b), and g'x=f(x)

2) Property:

∫baf(x)dx=-∫abf(x)dx

3) Given:

Fx=∫x1t+sin⁡t dt 

4) Calculation:

By using property,

∫x1t+sin⁡t dt=-∫1xt+sin⁡t dt

By using thefirst Fundamental Theorem of calculus

From the given function,

ft=t+sin⁡t

Replace t by x,

fx=x+sin⁡x

F'x=fx=-x+sin⁡x

Hence derivative of the given function is -x+sin⁡x

Conclusion:

Therefore,

F'x=-x+sin⁡x