23–68. Indefinite integrals Determine the following indefinite...

Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ ((2 + 3 cos y)/sin² y)dy

Accepted Answer

Welcome back, everyone. Evaluate the indefinite in role. In this problem, we went to integrate 5 + 4 cosine of X, all divided by sin squared of X. Before we begin the integration, we're going to split our fraction into two parts. For the first one, we take 5 divided by sine squared of X. We're going to write it as 5 multiplied by 1 divided by sine squared of x. And for the second one, we take 4 multiplied by cosine of X divided by sin squad of X. And what we're going to do is simply write sine squared of x as sign multiplied by sign. So we take 4 cosine of X divided by sin X, multiplied by 1 divided by sin x. So this is the same in a role that we want to evaluate. However, we can notice that we can apply trigonometric identities one divided by sine squared as cos squared of X. cosine divided by sine is cotangent, so we take cotangent of X, and one divided by sine of X is cosequent of X. So now we can rewrite our integral as the integral of 5, Cosequant squared of X plus 4 cotangent of X cosequant of X. Now we can split our integral into two parts and factor out the constants. So we get 5 multiplied by the integral of cosic and squad of X. Plus 4 multiplied by the integral of cotangent of X cosequent of X. And from here we can use the table of basic integrals. Let's remember that the derivative of cotangent is negative cosquad, right? But within our first in a grunt, we have cosequence squad with a positive sign. So what we're going to do is simply add a negative sign in front of 5. Followed by cotangents. Now we can check our work. The derivative of cotangent is negative cos squad, and negative multiplied by negative gives us a positive coefficient, meaning our first answer is correct. Now for the second one we are adding 4 multiplied by. Let's recall that the derivative of cosant is. Cos and cotangent. So once again, we need to add a negative sign. So we're multiplying 4 by negative cosequant of X. And finally, let's add the constant of integration C. And now we can write it in a cleaner form, so that'd be -5 cotangent of X minus 4 multiplied by cosequant of X plus C. That'll be our final answer, and thank you for watching.

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ ((2 + 3 cos y)/sin² y)dy

Key Concepts

Indefinite Integrals

Trigonometric Functions

Integration Techniques

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