Evaluate the integrals in Exercises 1–22.
∫₀^π 8 sin⁴(y)...

Question

Evaluate the integrals in Exercises 1–22.

∫₀^π 8 sin⁴(y) cos²(y) dy

Accepted Answer

Welcome back everyone. Compute integral from 0 to pi of 12 sin squared theta, cosine squared theta d theta. A 3 pi divided by 4, B 3 pi divided by 2. C pi divided by 4, and D pi divided by 8. So for this problem, let's go ahead and rewrite the integral using the basic properties of integrals. We can factor out to 12 because it's a constant and integrate from 0 to pi. Sine squad theta cosine squared theta d theta. Before we begin, let's analyze this expression. So using the properties of exponents, we can rewrite it as sine theta cosine theta squared, right? And from here, let's remember the double angle formula. If we take sin of 2 theta, this is going to be equal to. 2 sine theta cosine theta. If we divide both sides by 2, we're going to have 1/2 sin 2 theta equals sine theta cosine theta. And now if we square both sides. We are going to have one or. Sin squared to theta. This is how we can rewrite our integral, and basically if we multiply this expression by 12, we're going to have a constant of 3. So our integral is equal to 3, integral from 0 to pi of. Sin 22 theta d theta. Now from here we can simplify even further using the power reduction identity. Let's remember another identity, and that'll be sin squad x is equal to 1 minus. Cosine 2 x divided by 2. So in this case, we have sin 22 theta. Which is going to be equal to 1 minus cosine of or theta divided by 2. And now let's label the original integral with I. I is going to be equal to 3. We can factor out one half, and basically that'd be 3 divided by 2 integral from 0 to pi of. 1 minus cosine 4 theta d theta. Well done, and now let's go ahead and integrate. We have two separate integrals, the integral. Of 1 d theta, where basically d theta is going to be theta, so we end up with three halves, our constant in parent theta. And now minus the integral of cosine is sine. So we get sin of for theta, and according to the reverse chain rule, we have to divide by the slope of the linear term which is Or And finally we are going to evaluate the result from 0 to pi using the fundamental theorem of calculus Part 2. Which gives us three halves in parentheses. I Minus sign of 4 pi. Divided by 4? Minus in parentheses 0 minus sign of. 4 multiplied by 0 divided by 4. Let's remember that sin at the multiples of pi is going to be equal to 0, so we can cross out those sine terms and we basically end up with the only term which is pi multiplied by. The factor in front, which is we have and this gives us. We Pi divided by 2. Which corresponds to the answer choice b. Thank you for watching.

Evaluate the integrals in Exercises 1–22.∫₀^π 8 sin⁴(y) cos²(y) dy

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Evaluate the integrals in Exercises 1–22.
∫₀^π 8 sin⁴(y) cos²(y) dy