Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(...

Question

Evaluate the integrals in Exercises 1–22.

∫₀^(π/2) sin²(x) dx

Accepted Answer

Welcome back everyone. Evaluate the integral from 0 to pi of cosine squared XD X. A says pi divided by 2, B pi divided by 4, C, and D 0. For this problem, let's go ahead and rewrite the integral, that's the integral from 0 to pi of cosine squared XDX. To integrate cosine squared, we basically are going to use the half angle formula that will be the integral from 0 to pi. And let's remember that cosine squared x. Can be rewritten as 1 plus cosine of 2 x divided by 2. Now what we're going to do is simply factor out 1/2. Using the properties of integrals and integrate 1 plus cosine 2 x dx from 0 up to pi. And now let's go ahead and integrate. Specifically, we can split our integral into two parts. Let's write the constant. Let's add brackets. And then let's write two integrals. One of them is going to be the integral from 0 to pi of 1 dx or basically DX. And for the 2nd integral, we're going to have the integral from 0 to pi of cosine to x. D X. Now, let's continue the equation. That'll be 1/22. Now let's add brackets. The integral of the x is simply x, right? This is going to be evaluated from 0 up to pi, and then we want to add the integral of cosine to xdx from 0 to pi. So now, let's remember that the integral of cosine is equal to sine. And this is a composite function. So first of all we get sin of 2 x, and according to the reverse chain rule, we have to divide by the slope of the linear term. So we're going to divide by 2 and we get sin of 2 x divided by 2, evaluated from 0 to pi. Well done, so now let's apply the evaluation theorem, which gives us 1/2 brackets. X from 0 to pi gives us pi minus 0. And now we want to evaluate sine of 2 x divided by 2 from 0 to pi. We can factor out one half, and in parentheses we will have sin of 2 multiplied by pi. So that's sin of 2 pi. minus sign of 2 multiplied by 0, that's a sign of 0. So now sin at the multiple soft pi is 0, meaning we can cross out those terms. And we basically get 5 minus 0, which is pi multiplied by 1/2, that's pi divided by 2. Which corresponds to the answer of choice a. Thank you for watching.

Evaluate the integrals in Exercises 1–22.∫₀^(π/2) sin²(x) dx

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Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx