Evaluate the integrals in Exercises 31–56. Some integrals do n...

Question

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ sin(2x) cos(4x) dx

Accepted Answer

Welcome back everyone. Evaluate the integral sine 5 x, cosine 3 x d x. For this problem, let's recognize that we have a product of sine and cosine, right? So let's remember the formula. If we have sine of alpha multiplied by cosine of beta, we can rewrite it as a sum. Specifically, that would be one half in parentheses we would have sine of alpha plus beta. Plus sine of alpha minus beta. So in this case, our alpha is equal to 5 x. And beta is equal to 3 x. So we get 1/2 in parentheses. Sign of 5 x plus 3 x that's 8 x plus. Sign of alpha minus beta, so that'd be 5 x minus 3x, which is 2 x. So what we're integrating is. One half we can factor it out. Sign 8 x plus sin 2 xDX. We can now split it into two integrals, so we get one half integral of sin 8 x. Dx plus integral of sin. 2 XD X. Now let's go ahead and integrate. The integral of sign is. Negative cosine. So we get one half in parent, negative cosine of 8x, and according to the reverse chain rule we're dividing by the slope of the linear term. So in this case that would be divided by 8, and similarly, the integral of sine of 2 x is going to be negative cosine of 2 x divided by 2. And we're going to add a constant of integration c because that's an indefinite integral. And now let's distribute, so we're going to get -1 divided by 16 cosine of 8 x minus 1/4 cosine. Of 2 x plus c. And that's our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.∫ sin(2x) cos(4x) dx

Watch next

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ sin(2x) cos(4x) dx