Evaluating integrals Evaluate the following i...

Question

Evaluating integrals Evaluate the following integrals.

∫₀^²π cos² 𝓍/6 d𝓍

Accepted Answer

Welcome back, everyone. Evaluate the integral from 0 to 3 pi of cosine squared of C divided by 3 D Z. A says pi divided by 2, B 3 divided by 2, C pi divided by 3, and D 2 pi divided by 3. For this problem to integrate the square of cosine, we're going to use one of the trigonometric identities. Let's recall that cosine squared of X can be written as 1 plus cosine of 2 x divided by 2. So now, if we use this formula, we can rewrite cosine squared off C divided by 3S1 plus cosine of 2 Z divided by 3, all divided by 2. And now we can rewrite our integral as. Integral from 0 to 3 pi. Of 1 plus cosine of 2 z divided by 3. All divided by 2. And now, let's go ahead and split this integral into two parts, because we have a sum, right? So, the first integral is going to be the integral from 0 to 3 i of 1/2. We also want to include DEC, right, because we're integrating with respect to see, so we're integrating one half DEC. And for the second integral, we are integrating from 0 to 3 pi. Cosine of 2 z divided by 3. Divided by 2 DC. Now, what we're going to do is simply factor out each constant, so we can continue our equation, we are going to factor out one half. In C we're going to have two integrals, integral from 0 to 3 pi of DC. Plus integral from 0 to 3 pi of cosine of 2 z divided by 3 dz. Now, we can integrate the Z. The integral would be Z from 0 to 3 by, but for now, let's focus on cosine of 2 Z divided by 3. We're going to perform substitution. Let's suppose that U equals 2 Z. Divided by 3. And now our next step is to identify the derivative of U with respect to Z. Which is 2/3. From here, we can express D Z in terms of DU. D Z is going to be equal to DU multiplied by 3 divided by 2 or 3 halves DU. And now let's identify the new limits of integration. U1 is going to be equal to. 2 multiplied by 0, divided by 3, which is 0. And the upper limit of integration U2 is going to be equal to 2 multiplied by 3 pi. Divided by 3. And this is equal to 2 pi. So now we can rewrite our integral. Let's label it as I. I is equal to 1/2 in integral from 0 to 3 pi. Of DC Plus integral from 0 to 2 pi. Of cosine of you. Multiplied by DC which is 3 halves DU, so we're going to write DU and we can factor out three halves. And now we are ready to integrate, right? Because we can use tables of integrals to identify the value of each. We got one half in C Z evaluated from 0 to 3, that's the integral of D Z. Plus 3 halves. Of sign. Are for you Evaluated from 0 to 2 pi. Because the integral of cosine as sign, right? Now, let's perform the calculation. This is equal to 1/2 in Cs. We have Z from 0 to 3 pi. That's simply 3 pi minus 0, according to the fundamental theorem of calculus part 2. And then we add 3 halves in sign of 2 pi minus sign of 0. Since sign is 0 at the multiples of pi, we can say that sin of 2 pi is 0, sign of 0 is 0. So we simply end up with one half multiplied by 3 pi, which is 3 pi divided by 2. Well, once we have our answer now, considering the answer choices, we can conclude that the correct answer for this problem corresponds to the answer choice B.

Evaluating integrals Evaluate the following integrals.

∫₀^²π cos² 𝓍/6 d𝓍

Watch next

Evaluating integrals Evaluate the following integrals.∫₀^²π cos² 𝓍/6 d𝓍

Watch next

Evaluating integrals Evaluate the following integrals.

∫₀^²π cos² 𝓍/6 d𝓍