The integrals in Exercises 1–44 are in no particular order. Ev...

Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (csc t sin 3t dt)

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the integral, the integral ofcant of X multiplied by cosine of 3 XDX. OK. So it appears for this particular prom, we're ultimately asked to evaluate the integral that is provided to us by the prom itself. So with that in mind, our first step that we need to take is we need to rewritecant of X as 1 divided by cosine of X so that the integral will become the integral of cosine of 3 x divided by cosine of XDX. Our second step now is we need to recall and use the identity that cosine of 3 X is equal to or multiplied by cosine to the power of 3. X So once again, cosine of 3 X will be equal to 4 multiplied by cosine to the power of 3X minus 3, multiplied by cosine of X. To rewrite our numerator. So once again we're recalling and using the identity that allows us to write that cosine of 3 X is equal to 4 multiplied by cosine to the power of 3 x minus 3 multiplied by cosine of X. And our third step now. we need to recall and use our substitution method to help us write that the integral of 4 multiplied by cosine to the power of 3 X minus 3 multiplied by cosine of X. All divided by cosine of X D X will be equal to the integral of parenthesis 4 multiplied by cosine to the power of 2 X minus 3 DX. Fantastic. So, our fourth step now is we need to recall and use the identity that cosine to the power of 2 of X is equal to 1 plus cosine of 2 x divided by 2. And this will help us to rewrite. Or multiplied by cosine to the power of 2 of X. As for multiplied by parentheses 1 plus cosine of 2 x divided by 2, which will be equal to 2 multiplied by parentheses 1 plus cosine of 2 X, which will be all equal to 2 + 2 multiplied by cosine of 2 X. Fantastic. So With that in mind, we can go ahead and write that the integral of parentheses or multiplied by cosine to the power of 2 X minus 3 D X will be equal to the interval of parentheses of 2 + 2 multiplied by cosine of 2 X minus 3, and don't forget your parenthesis on the end, and this will be multiplied by DX. And this will be all equal to the interval of parentheses of 2 multiplied by cosine of 2 X minus 1. And don't forget your parenthesis in the end, multiplied by DX. Fantastic. We are on a roll here. So with that said, We can now begin our fifth and final step, which our final step is we need to integrate. So when we integrate, we will find that the integral of parentheses 2 multiplied by cosine of 2 X minus 1. In parenthesis multiplied by DX is equal to 2 multiplied by the integral of cosine of 2XDX minus the integral of DX which will be equal to. So once again, this will be all equal to. The integral of cosine of 2 X multiplied by D of 2 X. Which we will need to subtract. So now we need to subtract minus the integral of DX, which will be equal to sin of 2 X minus X plus C. Fantastic. And this will be all equal to sign of 2 X minus X plus C. Which we can also rearrange this expression to write that our final answer will also be equal to negative X plus of 2 X plus C, where C represents some constant value. And that's it. We've officially solved for this problem. So let's go look at the problem itself to recap what we've learned. So we've determined without a doubt that our evaluated integral is once again as follows. So our final answer without a doubt will be. Negative X plus sign of 2 X plus C. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫ (csc t sin 3t dt)

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The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (csc t sin 3t dt)