Evaluating the Integral of cos^n x over [0, 2π]: Traditional and Elementary Methods

When evaluating the integral of cosnx over the interval [0, 2π], we need to consider both traditional and elementary methods. Traditional methods often involve complex number manipulation and algebraic expansions, while elementary methods use reduction formulas to simplify the process. Let's explore both approaches in detail.

Traditional Method: Utilizing Complex Numbers and Binomial Expansions

The traditional approach involves expressing cosx in terms of the complex exponential function and expanding the expression using binomial coefficients. We start with the identity:

cos x frac{1}{2} ( z z-1 )

where z e iθ. For even n, we have:

cosnx left(frac{1}{2} ( z z-1 ) right)n

After expanding and simplifying, we get:

cosnx 2-n ( cos ( nx ) 2∑r1cos( n-2rx ) ? 2∑r1/2cos( 2x ) )

Observing that the integral of cosnx over [0, 2π] is zero for all n ≠ 0, we can simplify the integral as:

I -nCnπ-n[x]_0^{2π} 2{1-nCn/2n

Thus, for n 2020, the integral is:

I {1-2020C2020π}}{22020}/2020/1010!π

Calculating the exact value, we get:

I {-2019}C20201010π}}{22020}/1010!π≈0.1115.

Elementary Method: Applying the Reduction Formula

The elementary approach involves using the reduction formula:

I frac{1}{n} ( cosn-1xsinx frac{n-1}{n} I_n-2)

By repeatedly applying this formula, we can express the integral as:

I_n frac{(n-1)(n-3) ? 1}{n(n-2) ? 2} I_0

For n 2020, we have:

I_2020 frac{2019 ? 1}{2020 ? 2} I_0

Since the integral of cos over [0, 2π] is 2, we get:

I_2020 frac{2019 ? 1}{2020 ? 2} ? 2π frac{2020!}{2^{2020} ? 1010!^2} ? 2π frac{2π}{2^{2020}} ? C20201010

The result is approximately:

I_2020 ≈ 0.1115

Conclusion

Both the traditional and elementary methods provide a way to evaluate the integral of cosnx over [0, 2π]. The traditional method relies on complex numbers and binomial coefficients, while the elementary method uses a reduction formula. Each approach has its merits and can be applied depending on the context and the level of familiarity with the techniques involved.

Related Keywords

integral of cos^n x reduction formula binomial coefficient