Finding Antiderivatives
In Exercises 1–16, find an antid...

Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

(-3/2)csc²x(3x/2)

Accepted Answer

Welcome back everyone. Find an inside derivative of 8 divided by 5 co sequin squared of 5 divided by 8 x. For this problem we want to find the n inside derivative F x such that the derivative of FX is equal to the given function. So 8 divided by 5 coequin square of 5 divided by 8 multiplied by x. And what we can do is simply we call the basic derivatives. Let's remember that the derivative of cotangent of x is equal to negative co sequin squared x. Now if we apply the chain rule, let's suppose that we have a linear function cotangent of a linear function. So cotangent of a multiplied by x. Let's not introduce any y intercept because we don't have any. So the derivative of cotangent of a x is going to be equal to negative, a cosequent squared of a x according to the chain rule, right? And we're going to use this relationship to identify the required coefficients. Let's also multiply both sides by a constant C, so that C. Multiplied by cotencient. Of a x derivative is going to be equal to negative c coequin squared of a x. Now, what do we know? Well, essentially we know that A x in this case by analogy is equal to 5 divided by 8 multiplied by x. So from here we can immediately recognize that a is going to be equal to 5 divided by 8. What we want to do is identify the value of C, right? And what we can do is simply show that negative AC, our coefficient. Negative AC. is equal to 8. Divided by 5 because this is what we have in front of cosin squared and we can identify c which is 8 divided by 5 divided by a. Substituting the value of a, we're going to get -8. Divided by 5, which is multiplied by 5, divided by 8. And now let's go ahead and calculate the result. That's -64 divided by 25. So now what have we shown? Well, we have shown that 64 divided by 25 multiplied by cotangent. Of 5 divided by 8 multiplied by x derivative is equal to 8/5s co sequin squared off 5/8 x. So that's almost the answer that we need, but also let's remember that the derivative of a constant is zero, so we can identify the whole family of inside derivatives by adding a constant. We can show that -64 divided by 25. Cotangent of 5 divided by 8 multiplied by x plus some constant c, the derivative of this family is equal to 8 divided by 5 co c squared of. 5 divided by 8 multiplied by x. So now we have our family of an inside derivatives. Let's label it and thank you for watching.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.(-3/2)csc²x(3x/2)

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Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
(-3/2)csc²x(3x/2)