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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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[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.lfdahl said:Derive an expression for the definite integral:\[I = \int_{0}^{\frac{\pi}{4}}sec^m(x)dx, \;\;\;\;m = 2,4,6,...\]
lfdahl said:Hi, Opalg, thankyou for such a detailed and thorough answer!
Yes, I was asking for the solution with binomial expansion
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.
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.
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.
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.
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.