Why is Sinx X not integrable?
sinx/x is integrable only in the sense that you can integrate it from 0 to X and then let X become infinite. The integral from 0 to X exists in both Riemann and Lebesgue definitions.
Can Sinx X be integrated?
Answer : The expression on integrating sin x / x is x – (x3/3×3!)
What is the integration of sin x?
The integral of sin x is -cos x + C. It is mathematically written as ∫ sin x dx = -cos x + C.
Which functions Cannot integrate?
Some functions, such as sin(x2) , have antiderivatives that don’t have simple formulas involving a finite number of functions you are used to from precalculus (they do have antiderivatives, just no simple formulas for them). Their antiderivatives are not “elementary”.
Is Sinx lebesgue measurable?
The functions x↦sinx and x↦cos2x are continuous, hence Borel-measurable. Since sum, composites and products of Borel-measurable functions are again Borel-measurable, we conclude that f is Borel-measurable. Hence f is Lebesgue-measurable.
What is anti derivative of sinx?
cosx
A lot of people just memorize that the antiderivative of sinx is simply –cosx.
Can you integrate anything?
Not every function can be integrated. Some simple functions have anti-derivatives that cannot be expressed using the functions that we usually work with. One common example is ∫ex2dx.
Is it always possible to integrate a definite integral?
An indefinte integral cannot always be integrated analytically and may require numeric integration, while it is always possible to integrate a definite integral. Definite integrals always return a real number after evaluation at its limits of integration.
Is Sinx measurable?
These characteristic functions are Borel-measurable because both Q and R∖Q are Borel sets. The functions x↦sinx and x↦cos2x are continuous, hence Borel-measurable.
Is the sine integral from 0 to infinity elementary?
Today we have a tough integral: not only is this a special integral (the sine integral Si ( x )) but it also goes from 0 to infinity! Since this is a special integral, there is no elementary antiderivative and therefore we can’t simply plug the bounds into the result; this means none of the techniques we know of will work.
How to prove that f ( x ) is not integrable?
By the Limit Comparison Test applied to the series obtained above and the harmonic series, we conclude that the series diverges, hence ∫ 0 ∞ f + ( x) d x = ∞, so f ( x) is not integrable. x x. Surely the function is measurable, so we need to prove that
Is the Lebesgue integral of f + ( X ) integrable?
( x) / x. Since f + ( x) ⩾ 0, f + ( x) is integrable if and only if ∫ f + ( x) d x < ∞. Again, since f + ( x) ⩾ 0, we can compute its Lebesgue integral with the Monotone Convergence Theorem, and then estimate it from below: