Integrals: Fun to Solve Analytically!

In summary, the speaker has found two interesting integrals and was able to solve one analytically without the need for numerical solutions. They are offering to share the solution if no one else can solve it. The speaker also mentions the use of parametric integration as a recent learning method.
  • #1
LAZYANGEL
15
1
Hey folks,

found a couple of interesting integrals and was able to solve one of them ANALYTICALLY! That means no numerical solutions needed.

$$\int_{0}^{\frac{\pi}{2}} \frac{1}{1+(tan(x))^{\sqrt{2}}} dx$$

$$\int \frac{1}{1+e^{\frac{1}{x}}} dx$$

The first one I solved and will reveal analytic solution if no one can get it (it should be $$\frac{\pi}{4}$$).

Have fun!

To moderators: These are not homework or test problems, both of them are either from an old math olympiad and an old research paper.
 
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  • #2
Interesting integral, I tend to use my math handbook and just look these things up. However a method I learned recently was parametric integration. I don't know if that's what you used but I'll share a reference on it here in the interest of learning about new things:

http://www.maa.org/sites/default/files/268948443847.pdf
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the total value of a function over a specific interval.

2. Why are integrals important?

Integrals are important because they allow us to solve a wide range of problems in mathematics, physics, and engineering. They are also used to find the volume, centroid, and surface area of complex shapes.

3. What is the difference between a definite and indefinite integral?

A definite integral has specific limits of integration, while an indefinite integral does not. In other words, a definite integral has a defined start and end point, while an indefinite integral is a continuous function.

4. How can integrals be solved analytically?

Integrals can be solved analytically using various techniques, such as substitution, integration by parts, and trigonometric substitution. These techniques involve manipulating the integrand (the function being integrated) to make it easier to solve.

5. Are there real-world applications of integrals?

Yes, integrals have numerous real-world applications. They are used in physics to calculate the work done by a force, in engineering to find the center of mass of an object, and in economics to determine the total profit of a company. They also have applications in signal processing, statistics, and probability.

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