MHB Can You Solve This Definite Integral Challenge with Binomial Expansion?

AI Thread Summary
The discussion focuses on deriving the definite integral \( I = \int_{0}^{\frac{\pi}{4}} \sec^m(x) \, dx \) for even values of \( m \). The integral can be transformed using substitution and integration techniques, resulting in an expression involving Sloane's sequence A076729. Additionally, the binomial theorem is applied to express the integral as a sum, facilitating the calculation of the integral's value. Participants appreciate the innovative approach using binomial expansion. The conversation highlights the complexity and elegance of the mathematical solution.
lfdahl
Gold Member
MHB
Messages
747
Reaction score
0
Derive an expression for the definite integral:\[I = \int_{0}^{\frac{\pi}{4}}sec^m(x)dx, \;\;\;\;m = 2,4,6,...\]
 
Last edited:
Mathematics news on Phys.org
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]
 
Hi, Opalg, thankyou for such a detailed and thorough answer!:cool:

Yes, I was asking for the solution with binomial expansion
 
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
 
Last edited:
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top