67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the follo...

Question

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:

sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]

sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]

cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]

68. ∫ sin(5x)sin(7x) dx

Accepted Answer

Welcome back, everyone. Evaluate the integral of sin of 3X, sign of 8XDX using the product to some identities. For this problem we have a product of two signs. Let's recall that sin of alpha multiplied by sin of beta can be rewritten as a sum. We begin with 1/2 and in C we have cosine of alpha minus beta. Minus cosine of alpha plus beta. So let's convert sin of 3 X of 8 X into a sum. That'd be 1/2 in, cosine of 3 x minus. 8 X is -5X, and we're subtracting cosine of 3 X plus 8X which is E11 X. Meaning our integral becomes integral of one half we can factor out 1/2, right? And within our integrat, we have cosine of -5 X, which is equal to. Cosine of positive 5 x because cosine is an even function. So we take cosine of 5 X and subtract cosine of 11 X. Followed by the X. So now let's split into two integrals. We're going to factor out the constant, 1/2. And we want to integrate cosine. Of 5 XD X and subtract the integral of cosine. Of 11 X D X And now let's value the result. We got 1/2. The integral of cosine is sign, so we first will get sine of 5 X. And let's recall that integrating cosine of a linear function, such as 5 X. Requires us to divide by the slope of the linear functions, we also divide by 5, meaning we get sin of 5 X divided by 5. And we apply the same concept for the next integral. Integral of cosine of 11 X is going to be equal to sine of 11 X. Divided by 11. And finally, let's add a constant of integration C because we have an indefinite integral. And now we can distribute, so that would be sign. Of 5 Xs divided by 10. Minus sine of 11 X divided by 22. Plus a constant of integration C. Which is our final answer for this problem, and thank you for watching.

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
68. ∫ sin(5x)sin(7x) dx

Key Concepts

Product-to-Sum Identities

Integration Techniques

Definite vs. Indefinite Integrals

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