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lfdahl
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Calculate the following definite trigonometric integral:
\[\int_{0}^{\frac{\pi}{2}} \cos^{2017}x \sin^{2017}x dx\].
\[\int_{0}^{\frac{\pi}{2}} \cos^{2017}x \sin^{2017}x dx\].
my solution :lfdahl said:Calculate the following definite trigonometric integral:
\[\int_{0}^{\frac{\pi}{2}} \cos^{2017}x \sin^{2017}x dx\]=A.
Thankyou, Albert!, for your fine solution! Well done.Albert said:my solution :
$Using \,Beta \,Function :$
$$\beta(m,n)=2\int_{0}^{\frac{\pi}{2}}sin^{2m-1}x\,\,cos^{2n-1}x\, dx
=\dfrac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}=\dfrac{(m-1)!(n-1)!}{(m+n-1)!}$$
here $m=n=1009$
so $A=\dfrac{\Gamma(1009)\Gamma(1009)}{2\Gamma(2018)}=\dfrac{(1008)!\times(1008)!}{2\times(2017)!}$
A definite integral is a mathematical concept that represents the area under a curve between two specific points on a graph. It is denoted by the symbol ∫ and is used to find the total value of a function over a given interval.
The definite integral of a function is calculated using a process called integration. This involves finding the anti-derivative of the function and then evaluating it at the upper and lower limits of the given interval.
The number 2017 is an arbitrary constant in the given function and does not hold any particular significance. It is simply used to represent a specific value in the function and can be replaced with any other number without changing the overall concept of the function.
The definite integral challenge ∫cos2017xsin2017xdx can be solved by first finding the anti-derivative of the function, which in this case is (1/2)sin(4034x). Then, the anti-derivative is evaluated at the upper and lower limits of the given interval to find the difference between the two values.
Definite integrals have many real-world applications, such as finding the total distance traveled by an object with varying velocity, calculating the total amount of work done by a force, and determining the total amount of revenue generated by a business over a given time period. They are also used in physics, engineering, and economics to solve complex problems involving rate of change and accumulation.