Evaluate the integrals in Exercises 69–134. The integrals are ...

Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (2 − cosx + sinx) / sin²x dx

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 of 1 + 2 multiplied by sin of x minus cosine of x, all divided by sin square of x dx. OK. So, it appears for this particular prong, we're given an integral and we're ultimately asked to evaluate this integral. So now that we know that we're ultimately trying to evaluate this integral. Our first step that we need to take, based on our prior knowledge, is that we need to split the integrand into simpler terms. So what is the integrand? Well, the integrand for this problem is 1 + 2 multiplied by sin of x minus cosine of x, all divided by sine to the power of 2 of x. And this will be equal to once again we're splitting our integrand into simpler terms so we can determine that this will be equal to 1 divided by sin squared of x plus 2 multiplied by sin of x divided by sine to the power of 2 of x minus cosine. Of x divided by sin to the power of 2 of x, and this will be all equal to cosicant squared of x plus 2 multiplied by cosicant of x minus cotangent of x multiplied by cosicant of x. Fantastic. So our 2nd step now, so this will be step 2. Is we need to integrate term wise and we also need to recall our standard antiderivatives. So when we do that, we'll be able to write that the integral of cosequant square of x dx will be equal to negative cotangent of x plus C subscript 1. You can also write that the integral of cosecant of x dx is equal to negative natural log of the absolute value of cosecant of x plus. Cotangent of X. Plus C subscript 2. And finally, we can write the integral of cotangent of X, so cotangent of X multiplied by cosecant of X. DX will be equal to negative coequant of X plus C subscript 3. Fantastic. So with that in mind. Our third step is we need to combine our results, so we can go ahead and write that the integral of 1 + 2 multiplied by sin of x minus cosine of x, all divided by sin squared of x dx will be equal to the integral of. Cosequant squared of xDX plus 2 multiplied by the integral of cosequant of xDX. Minus the integral of cotangent of x multiplied by cosecant of x dx fantastic. And this will be equal to negative cotangent of x minus 2 multiplied by the natural log of the absolute value of cosequant of x plus cotangent of x minus parentheses of negative coequant of x plus c. Which we should be able to take a moment here and recall that this variable C denotes a constant variable, so we need to be able to recall that piece of information. So for our 4th and final step is we now need to simplify, and when we simplify, we will find that the integral of 1 + 2 multiplied by sin of x minus cosine of x, all divided by sin squared of x. D X Will be equal to. Poseant of x minus cotangent of x minus 2 multiplied by the natural log of the absolute value of coequant of x plus cotangent of x. Plus C and that's it. That is our final answer. Hooray, we did it. So going back to the talk to recap what we've learned, we can safely say that our final answer for our evaluated integral will once again be, so saying it once. Again, cosicant of x minus cotangent of x minus 2 multiplied by the natural log of the absolute value of cosicant of x plus cotangent of x plus c. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.∫ (2 − cosx + sinx) / sin²x dx

Watch next

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (2 − cosx + sinx) / sin²x dx