# A sinusoid integrated from -infinity to infinity

• pennyantics
In summary, the conversation is discussing the integral of sin(x)^3 dx over all real numbers and how the function's antisymmetry leads to the assumption that the integral equals zero. A possible proof using the Cauchy Principal Value is also mentioned, but it is noted that strictly speaking the integral does not exist.

#### pennyantics

I had a sort of odd question on my homework,

Sin(x)^3 dx, integrated over all reals (from negative infinity to infinity).

The problem also gives this morsel of ambiguity:

"Hint: think before integrating. this is easy"

Now my initial guess because of the antisymmetry of the function is that it equals zero. Although the problem doesn't ask for a proof of any way shape or form however, I was baffled how I would argue that I reasoned it equaled zero if I was called upon in class.

So I'm wondering whether my assumption is correct as well as maybe a brief explanation. No proof needed.

pennyantics said:
Now my initial guess because of the antisymmetry of the function is that it equals zero.

Exactly correct.

A pedant might ask for proof that you can use the antisymmetry of the integral in this way when the limits are +- infinity.

But I guess that's why we have mathematicians.

$$\lim_{a\to\infty}( \int^a_{-a} \sin^3 x dx ) = \lim_{a\to\infty} (\int^a_0 \sin^3 x dx + \int^0_{-a} \sin^3 x dx) =\lim_{a\to\infty} (F(a) - F(0)) - (F(0) + F(-a))$$, Where dF(x)/dx=sin^3 x.

Since the derivative of any odd function is an even function, F(-a)=F(a)

$$\lim_{a\to\infty} (F(a) - F(0)) - (F(0) + F(-a)) = \lim_{a\to\infty} (F(a) - F(0)) - (F(0) + F(a)) =\lim_{a\to\infty} (0) = 0$$.

What Gib Z gives is the "Cauchy Principal Value" of the integral. Of course, the limit is 0 because sin(x) is an odd function. Evaluating its integral at a and -a will give the same thing.

Strictly speaking $\int_{-\infty}^\infty f(x)dx$ is
$$\lim_{a\rightarrow -\infty}\int_a^0 f(x)dx+ \lim_{b\rightarrow \infty} \int_0^b f(x) dx$$
where the two limits are taken independently. Using that definition,
$$\int_{-\infty}^\infty sin^3(x) dx$$
does not exist.

## 1. What is a sinusoid integrated from -infinity to infinity?

A sinusoid integrated from -infinity to infinity is a mathematical concept that represents the total area under a sinusoidal curve from negative infinity to positive infinity. It is also known as the "indefinite integral" of a sinusoidal function.

## 2. How is the integral of a sinusoid calculated?

The integral of a sinusoid can be calculated using the fundamental theorem of calculus, which states that the integral of a function is equal to the area under the curve of the function. In the case of a sinusoid integrated from -infinity to infinity, the integral is calculated using the formula: ∫ f(x) dx = -cos(x) + C.

## 3. What is the significance of integrating a sinusoid from -infinity to infinity?

The integration of a sinusoid from -infinity to infinity allows us to find the total area under the curve of a sinusoidal function, which is useful in many mathematical and scientific applications. It also helps in solving differential equations and understanding the behavior of periodic functions.

## 4. Can a sinusoid integrated from -infinity to infinity have a finite value?

Yes, a sinusoid integrated from -infinity to infinity can have a finite value if the function is bounded and has a finite amplitude. For example, the integral of sin(x) from -infinity to infinity is equal to 0, as the function is bounded between -1 and 1.

## 5. How is a sinusoid integrated from -infinity to infinity used in real-world applications?

The concept of integrating a sinusoid from -infinity to infinity is used in various fields such as physics, engineering, and signal processing. It is used to analyze and model the behavior of periodic phenomena, such as sound waves, electrical signals, and mechanical vibrations. It is also used in calculating the area under the curve of a function, which has numerous practical applications in fields such as economics, statistics, and finance.