Evaluate the integrals in Exercises 53–58.
∫ from 0 to π...

Question

Evaluate the integrals in Exercises 53–58.

∫ from 0 to π/2 of sin(x) cos(x) dx

Accepted Answer

Welcome back everyone. Evaluate the definite integral from 0 to pi divided by 4 of sin x, cosine x, d x ass 0 B 1/22, C 14, and D 18. Before we begin this problem, let's simplify the function that we're integrating. Let's remember the double angle identity. Specifically, sine of 2 alpha is equal to 2 sine alpha, cosine alpha. So if we divide both sides by 2, we are going to show that 1/2 sin of 2 alpha is equal to sine alpha. Cosine alpha and now what does that mean? Well, it essentially means that we can rewrite our integral as integral from 0 to pi divided by 4. Of 1/2 sign of 2 x. DX. We can now factor out the constant, so that would be one half integral from 0 to pi divided by 4 of sin of 2 x d x. Now what we can do is simply use U substitution. Let's suppose that U is equal to 2 x. The derivative of you with respect to x is the derivative of 2x, which is 2. And now DX is going to be equal to DU divided by 2, or basically 1 half DU. We can change the limits of integration. The lower one is u1 equals 2 multiplied by 0, which is 0, and the upper one is u2 equals 2 multiplied by pi divided by 4, which is 2 pi divided by 4, or pi divided by 2. This allows us to rewrite the integral as 1/2. Integral from 0 to pi divided by 2. Of sin of u multiplied by dx, which is 1/2 du u, so we can write duu and we can factor out 1/2, and 1/2 multiplied by 1/22 is 1/4. Well done, so now we have an elementary integral. The integral of sin of u is equal to negative cosine, so we get 1/4 multiplied by negative. So we can just write -1/4 cosine of u. Evaluated from 0 to pi divided by 2. Now let's go ahead and factor out 1/4 and apply the fundamental theorem of calculus part 2. So we get cosine of pi divided by 2. Minus cosine of 0. Cosine of pi divided by 2 is equal to 0. And cosine of zero is equal to 1, so we get -1 multiplied by -14, which is 1/4. And therefore the correct answer is option C. Thank you for watching.

Evaluate the integrals in Exercises 53–58.∫ from 0 to π/2 of sin(x) cos(x) dx

Watch next

Evaluate the integrals in Exercises 53–58.
∫ from 0 to π/2 of sin(x) cos(x) dx