Can You Solve This Definite Integral Challenge with Binomial Expansion?

In summary, the expression for the definite integral is $\int_0^1(1+t^2)^n\,dt = \sum_{k=0}^n\frac1{2k+1}{n\choose k}.$ This integral can also be expressed in terms of a hypergeometric function, but the sequence A076729 provides a simpler representation. The value of the integral is $\dfrac{a(n)}{1\cdot3\cdot5\cdots(2n+1)},$ where $a(n)$ is the $n$th term in the sequence.
  • #1
lfdahl
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Derive an expression for the definite integral:\[I = \int_{0}^{\frac{\pi}{4}}sec^m(x)dx, \;\;\;\;m = 2,4,6,...\]
 
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  • #2
lfdahl said:
Derive an expression for the definite integral:\[I = \int_{0}^{\frac{\pi}{4}}sec^m(x)dx, \;\;\;\;m = 2,4,6,...\]
[sp]\[I = \int_0^{\pi/4}\sec^{m-2}x\sec^2x\,dx = \int_0^{\pi/4}\sec^{m-2}x\,d(\tan x) = \int_0^1(1+t^2)^n\,dt,\] where $t = \tan x$ and $n = (m-2)/2.$ The value of this integral is $\dfrac{a(n)}{1\cdot3\cdot5\cdots(2n+1)},$ where $a(n)$ is the $n$th term in Sloane's sequence A076729. This sequence can be expressed in terms of hypergeometric functions, but not in any simpler way.

Another way to express the answer would be to use the binomial theorem to write the integral as \[\int_0^1(1+t^2)^n\,dt = \int_0^1\sum_{k=0}^n{n\choose k} t^{2k}dt = \sum_{k=0}^n{n\choose k}\int_0^1 t^{2k}dt = \sum_{k=0}^n\frac1{2k+1}{n\choose k}.\]
[/sp]
 
  • #3
Hi, Opalg, thankyou for such a detailed and thorough answer!:cool:

Yes, I was asking for the solution with binomial expansion
 
  • #4
lfdahl said:
Hi, Opalg, thankyou for such a detailed and thorough answer!:cool:
Yes, I was asking for the solution with binomial expansion
Opalg's solution with binomial expansion:
Another way to express the answer would be to use the binomial theorem to write the integral as \[\int_0^1(1+t^2)^n\,dt = \int_0^1\sum_{k=0}^n{n\choose k} t^{2k}dt = \sum_{k=0}^n{n\choose k}\int_0^1 t^{2k}dt = \sum_{k=0}^n\frac1{2k+1}{n\choose k}.\]
Innovative ! I like the solution with binomial expansion
 
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FAQ: Can You Solve This Definite Integral Challenge with Binomial Expansion?

What is the "Definite Integral Challenge"?

The Definite Integral Challenge is a mathematical problem that involves calculating the area under a curve between two points. It is often used to solve real-world problems such as finding the distance traveled by an object or the total amount of a substance in a given volume.

What is a definite integral?

A definite integral is a mathematical concept that represents the area under a curve between two points. It is calculated by dividing the area into smaller rectangles and then adding up the areas of each rectangle. The smaller the rectangles, the more accurate the calculation will be.

How is the area under a curve calculated?

The area under a curve is calculated using a definite integral. The integral is essentially a sum of infinitesimally small rectangles under the curve. The smaller the rectangles, the more accurate the calculation will be. The integral is represented by the symbol ∫ and is often accompanied by the function and the limits of integration.

What are the applications of definite integrals?

Definite integrals have many real-world applications, such as finding the area under a curve, calculating volumes of irregular shapes, and determining the average value of a function. They are also used in physics, engineering, and economics to solve problems involving rates of change and accumulation.

How can I improve my skills in solving definite integral challenges?

The best way to improve your skills in solving definite integral challenges is to practice regularly. Start with simple problems and gradually increase the level of difficulty. You can also seek help from online resources, textbooks, or a tutor. It's also important to have a strong understanding of basic calculus concepts, such as derivatives and antiderivatives, before attempting to solve definite integral challenges.

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