# Quite complicated integral= sin(π/12)

• MHB
• Tony1
In summary: Simplifying further, we get:$$\sqrt{3}\int_{0}^{\infty}{\frac{e^{i\pi\frac{u^2}{3}}+e^{-i\pi\frac{u^2}{3}}}{e^{u\pi}+e^{-u\pi}}}\mathrm du$$Using the property of complex conjugates again, we can rewrite the integral as:$$Tony1 Given that, How to show that,$$\int_{0}^{\infty}{\cos(\pi x^2)\over {1\over 2}+\cosh\left({x\pi\over \sqrt{3}}\right)}\mathrm dx=\sin\left(\pi\over 12\right)$$Hello! To show that the given integral is equal to \sin\left(\frac{\pi}{12}\right), we can use the substitution u=\frac{x}{\sqrt{3}} to rewrite the integral as:$$\int_{0}^{\infty}{\cos(\pi x^2)\over {1\over 2}+\cosh\left({x\pi\over \sqrt{3}}\right)}\mathrm dx=\sqrt{3}\int_{0}^{\infty}{\cos\left(\pi\frac{u^2}{3}\right)\over {1\over 2}+\cosh(u\pi)}\mathrm du$$Next, we can use the identity \cosh(x)=\frac{e^x+e^{-x}}{2} to simplify the denominator:$$\int_{0}^{\infty}{\cos\left(\pi\frac{u^2}{3}\right)\over {1\over 2}+\cosh(u\pi)}\mathrm du=\sqrt{3}\int_{0}^{\infty}{\cos\left(\pi\frac{u^2}{3}\right)\over \frac{1}{2}+\frac{e^{u\pi}+e^{-u\pi}}{2}}\mathrm du$$Using the trigonometric identity \cos(x)=\frac{e^{ix}+e^{-ix}}{2}, we can rewrite the numerator as:$$\cos\left(\pi\frac{u^2}{3}\right)=\frac{e^{i\pi\frac{u^2}{3}}+e^{-i\pi\frac{u^2}{3}}}{2}$$Substituting this into the integral, we get:$$\sqrt{3}\int_{0}^{\infty}{\frac{e^{i\pi\frac{u^2}{3}}+e^{-i\pi\frac{u^2}{3}}}{2}\over \frac{1}{2}+\frac{e^{u\pi}+e^{-u\pi}}{2}}\mathrm du$$Using the property of complex conjugates, we can rewrite the integral as:$$\sqrt{3}\int_{0}^{\infty}{

## 1. What is a complicated integral?

A complicated integral is a mathematical expression that involves finding the area under a curve using advanced techniques. It requires knowledge of calculus and can involve complex functions and limits.

## 2. What is the value of the integral sin(π/12)?

The value of the integral sin(π/12) is approximately 0.2588. This can be calculated using trigonometric identities and integration techniques.

## 3. Why is the integral sin(π/12) considered to be quite complicated?

The integral sin(π/12) is considered quite complicated because it involves a trigonometric function, which can be difficult to integrate. Additionally, the value of π/12 is not a commonly used angle measure, making it less familiar to work with.

## 4. What applications does the integral sin(π/12) have in science?

The integral sin(π/12) has various applications in fields such as physics, engineering, and mathematics. It can be used to model wave motion, calculate areas and volumes, and solve differential equations.

## 5. Can the integral sin(π/12) be simplified or approximated?

Yes, the integral sin(π/12) can be simplified or approximated using various techniques such as substitution, trigonometric identities, or numerical methods. However, the exact value of the integral is a decimal and cannot be simplified to a simple fraction or whole number.

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