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.

∫ sin³(θ) cos(2θ) dθ

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Hello. In this video we are going to be evaluating the following integral. The integral given to us is the integral of sine to the 5th power of x multiplied by cosine of 2 x dx. Now what we want to do is we want to try to simplify this integral by using trigonometric identities. Let's go ahead and start by simplifying cosine of 2x. Now we can simplify cosine of 2x by using the double angle identity. Cosine of 2 x is equivalent to 2 cosines squared of x. -1. So if we were to apply this identity. We can rewrite the integral to be the integral of sine to the 5th power of x multiplied by the quantity 2 cosines squared of x minus 1 DX. Now the next thing we want to simplify is you want to simplify sine to the 5th power of x. Now because this is a really high exponential value of sine, what we want to do is we want to try to rewrite it into products of sine that use an exponent of 2. That way we can be able to apply trigonometric identities. So the first thing we can do is we can separate this to be sin of x multiplied by sin to the 4th power of x. Sine to the 4th power of x is also equivalent to sin squared quantity squared, so we can further rewrite this as sin of x multiplied by the quantity sin squared of x raised to the power of 2. So, if we were to apply this substitution, we can further rewrite the integral to be the integral of sin of x multiplied by the quantity sin squared of x raised to the power of 2 multiplied by 2 cosine of x, 2 cosine square of x minus 1 dx. Now, we can further simplify this by applying a trigonometric identity to sin squad of x. Sin square x is equivalent to 1 minus cosine square of x. Further applying this identity allows us to rewrite the integral to be the integral of sin of x multiplied by the quantity 1 minus cosine squared of x raised to the power of 2. Multiplied by 2 cosines squared of x minus 1 d x. Now, at this point, what we can go ahead and do is apply a use substitution to further simplify this integral. Let's go ahead and allow you to equal the cosine of X. Then DU is going to equal to negative sin of x dx, which can further be rewritten as duu is equal to sin of x dx, which we do have in the integral. So by applying our U substitution, we can further simplify this integral to be the integral of the quantity 1 minus u2 quantity 2 multiplied by 2U2 minus 1 multiplied by negative DU. And moving the negative sign to the front of the integral gives us the negative integral of the quantity 1 minus u2 quantity squared multiplied by 2 u 2. -1 DU So, let's go ahead and start by expanding 1 minus U2 quantity 2. That will allow us to rewrite the integral to be the negative integral of the quantity 1 minus 2U2 plus u 4 multiplied by 2U2 minus 1DU. Then foiling these quantities together leaves us with the negative integral of 2 u to the 6th. -5 u to the 4th plus 4 u2 minus 1 d u. At this point here, we can go ahead and perform term by term integration. That will leave us with -1 multiplied by 2/7. You to the 7th minus u to the 5th. Plus 4/3 u 3 minus u + c. Then if we were to distribute the -1 to each term, we get -2/7 u to the 7 plus u to the 5th minus 4/3 u 3 plus u plus c. Now at this point here, we want to go ahead and redistribute back our original use substitution of cosine of x. So this will give us a final solution of -2/7 cosine to the 7th power of x plus cosine to the 5th power of x minus 4/3 cosine cubed of X plus cosine of x plus C. This is going to be the final solution to the given integral. And the overall solution to this problem is going to be option A. 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.∫ sin³(θ) cos(2θ) dθ

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Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin³(θ) cos(2θ) dθ