# Simple closed form for integral

• MHB
• Tony1
In summary, a simple closed form for an integral is an expression that represents the exact value of an integral without the need for further calculations. It is important because it allows for efficient and accurate computation of integrals, and can be derived using various mathematical techniques. However, not all integrals have a closed form solution and require numerical methods for approximation. In real-world applications, a simple closed form for an integral can be used for important calculations and to solve differential equations.
Tony1
How may we go about to show that,

$$\int_{0}^{1}t\cos(2t\pi)\tan(t\pi)\ln[\sin(t\pi)]\mathrm dt=\color{green}{1\over \pi}\cdot\color{blue}{{\ln 2\over 2}(1-\ln 2)}$$

A hint is requested ... (Blush)

lfdahl said:
A hint is requested ... (Blush)
can I get the hint?

64 days after my 1st request:

A hint is still requested ... (Wave)

## 1. What is a simple closed form for an integral?

A simple closed form for an integral is an expression that represents the exact value of an integral without the need for further calculations. It typically involves elementary functions such as polynomials, trigonometric functions, and exponentials.

## 2. Why is a simple closed form for an integral important?

A simple closed form for an integral is important because it allows for efficient and accurate computation of integrals, which are fundamental in many areas of science and engineering. It also provides a better understanding of the behavior of functions.

## 3. How is a simple closed form for an integral derived?

A simple closed form for an integral can be derived using various techniques such as substitution, integration by parts, and trigonometric identities. It often requires a deep understanding of calculus and advanced mathematical concepts.

## 4. Can all integrals be expressed in a simple closed form?

No, not all integrals can be expressed in a simple closed form. In fact, many integrals do not have a closed form solution and require numerical methods for approximation. This is especially true for more complex integrals involving special functions.

## 5. How can a simple closed form for an integral be used in real-world applications?

A simple closed form for an integral can be used in various real-world applications, such as physics, engineering, economics, and statistics. It allows for the calculation of important quantities such as area, volume, center of mass, and probability. It also helps in solving differential equations and modeling real-world phenomena.

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