Exercises 59–64 require the use of various trigonometric ident...

Question

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.

∫ cos²(2θ) sin(θ) dθ

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Hello. In this video, we are going to be computing the following integral. Now we are given the integral of cosine squared of 4 t multiplied by sine of 2 t dt. Now, the first thing we want to do is you want to try to simplify the highest power trigonometric function in the integral, which is going to be cosine squared of 4 T. Now what we can go ahead and do in this case is we can use the double angle identity to simplify the cosine function. Now recall, at least for this case, the double angle identity is cosine of 2 theta is equal to 2 cosine squared theta. -1. But here we are given the angle of 4 T. So if we allow theta to equal to 2T, then we can rewrite the identity to be cosine of 4 T. is equal to 2 cosine squared of 2 t minus 1. Now if we were to apply this identity to the given integral, we can rewrite the integral as the integral of the quantity to cosine squared of 2 t minus 1, all raised to the power of 2 multiplied by sine of 2 t dt. Now here, what we can go ahead and do is we can apply a U substitution to further simplify the integral. If we allow u to equal the cosine of 2 t, then d u is equal to negative sin of 2 t multiplied by 2. And this can further simplify to be negative 1/2 d u is equal to sin of 2 t dt which we have in the integral. So, by applying this U substitution, we can further simplify the integral to be the integral of 2 U 2 minus 1 quantity 2 multiplied by 1/2DU. If we were to move the coefficient to the front of the integral, we will have -1/2 multiplied by the integral of the quantity 2u2 minus 1 quantity squad DU. Now, if we were to expand the quantity inside the integrand, we can further rewrite the integral to be -1/2 multiplied by the integral of 4U 4 minus 4U2 + 1DU. Finally, by doing term by term integration, we get 1/2 multiplied by 4/5 u to the 5th. -4/3 u 3 plus u + c. We will then distribute the coefficient of -1/22. To each term. That will leave us with -2/5 u to the 5th. Plus 2/3 u/3 minus 1/2 u plus c. And finally, if we backs substitute our original use substitution, the final solution to this problem is going to be -2/5, cosine to the 5th of 2t plus 2/3 multiplied by cosine to the 3 of 2 t minus 1/2 multiplied by cosine of 2 t plus c. This is going to be the solution to the given integral, and the overall solution to this problem is going to be option C. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.∫ cos²(2θ) sin(θ) dθ

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Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ cos²(2θ) sin(θ) dθ