{Use of Tech} Powers of sine and cosine It can be shown that

Question

∫ from 0 to π/2 of sinⁿx dx = ∫ from 0 to π/2 of cosⁿx dx =

{

[1·3·5···(n-1)]/[2·4·6···n] · π/2 if n ≥ 2 is even

[2·4·6···(n-1)]/[3·5···n] if n ≥ 3 is odd

}

b. Evaluate the integrals with n = 10 and confirm the result.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This process is to evaluate the integral from 0 to pi divided by 2 of cosine to the 4th of XDX and confirm whether it matches the given formula for N equal to 4. And we're given the formula that the integral from 0 to pi divided by 2 of cosine to the end of X. DX is equal to the quantity of 1 multiplied by 3, multiplied by 5, all the way up to the quantity of N minus 1 in quantity, and that's going to be divided by. The quantity of 2 multiplied by 4 multiplied by 6, all the way up to N in quantity multiplied by pi divided by 2, if N greater than equal to 2 is an even integer. So the first step in this problem is just to evaluate our integral, and so we're given the integral from 0 to pi divided by 2 of cosine to the 4th of XDX. So we're going to need to rewrite this integral using some trig identities. So this is going to be equal to the integral from 0 to pi divided by 2, and here we're going to rewrite. Are cosine to the 4th of x as the quantity of cosine squared of X in quantity squared X. So now we're going to use some trigger identities to rewrite cosine squared of X. So this is going to be equal to the interval from 0 to pi divided by 2. Of the quantity, and here for cosine squared of x, recall that that's going to be the quantity of 1 plus cosine of 2 X in quantity, divided by 2, in quantity squared DX. And so now we're going to do is multiply out our integrand. In doing so, this is going to be equal to the integral from 0 to pi divided by 2. Of 1 divided by 4, multiplied by the quantity of 1 plus. 2 multiplied by cosine of 2 X. Plus, Cosine squared of 2 X. In quantity DX. And now finally we'll just rewrite that cosine squared of 2 x in terms of cosine of 4 X. So in doing so, this is going to be equal to the integral from 0 to pi divided by 2 of 1 divided by 4, multiplied by the quantity of 1 + 2 multiplied by cosine of 2 X. Plus the quantity of 1 plus cosine of 4 X in quantity, divided by 2 in quantity DX. And so finally, just collecting like terms, we see that our intervals is going to be equal to. The interval from 0 to pi divided by 2 of 1 divided by 4 multiplied by the quantity, and here we have 1 plus 1 divided by 2, which is going to give us 3 divided by 2, and we'll have plus 2 multiplied by cosine of 2 X. Plus 1 divided by 2 multiplied by cosine of 4 X in quantity DX. So now all we need to do is to integrate each of these terms individually. In doing so, we'll see that our integral is going to be equal to 1 divided by 4 multiplied by the quantity, and here we're going to be integrating the constant of 3 divided by 2, and recall that when we integrate a constant, we get 3 divided by 2 multiplied by x. So we just get that constant back multiplied by x. Then we'll have plus, and here for the next term we're integrating 2 multiplied by cosine of 2 X. and when we integrate 2 multiplied by cosine of 2 X with respect to X, we get sine. Of 2 X. They will have plus and for the last term. We will have One half, multiplied by the cosine of 4 X being integrated with respect to X, and we do so, we get 1 divided by 8, multiplied by sine of 4 X. And that whole quantity is going to be evaluated from 0 to pi divided by 2. So now, substituting in our limits of integration, we see that this is going to be equal to. 1 divided by 4, multiplied by the quantity of 3 divided by 2, multiplied by pi divided by 2. Plus the sign of 2 multiplied by pi divided by 2. Plus 1 divided by 8 multiplied by the sine of the quantity of 4 multiplied by pi divided by 2 in quantity. Then we'll have minus the quantity for a lower limit. We'll have 3 divided by 2 multiplied by 0, plus the sign of the quantity of 2 multiplied by 0. In quantity plus 1 divided by 8, multiplied by the sign of the quantity of 4, multiplied by 0, in quantity. And so simplifying this expression, we see that this is going to be equal to. 3 pi divided by 16. Plus 0. Plus 0 minus 0-0-0. Since the sign of pi and the sign of 2 pi are equal to 0, and the sign of zero is also equal to 0. So it means that this integral is just going to be equal to 3 pi divided by 16. So that is the answer that we're looking for for the evaluation of the integral. So now we need to see if it matches the formula that we were given in the problem. And so using the formula that we're given the problem, we see that the integral from 0 to pi divided by 2 of cosine to the 4th of X D X is going to be equal to, and here in the numerator we need to multiply together the odd integers going from 1 ton minus 1. And N minus 1 in this case is going to be 4 minus 1, which is 3. So in the numerator we'll just have 1 multiplied by 3. And this quantity is going to be divided by the quantity of the even integers multiplied together going from 2 ton, and here in this case n is equal to 4, so we'll just have 2 multiplied by 4. And then this fraction gets multiplied by pi divided by 2. And when we evaluate this expression, we see that this is going to be equal to 3 pi divided by 16. So that is the same answer that we got when we evaluate the the expression, and so that means that that formula does match the evaluation of our integral. So this here is going to be the answer to our problem, which is going to be 3 pi divided by 16. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Definite Integrals

Properties of Sine and Cosine Functions

Gamma Function and Factorials

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