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.

−π csc (πx/2) cot (πx/2)

Accepted Answer

Welcome back, everyone. Find the and derivative of negative 5 halves cosequent of 5 halves x cotangent of 5 halves X. A says -25 divided by 4 cosequent of 5 halves X plus C. B cosequent of 5 halves X plus C. C negative cosequent 5 halves X plus C. And the cosequant of 2/5 X plus C. For this problem, let's understand that if we begin with a function F of X and we find it's derivative to return a function lowercase f x, then F of X is the antiderivative of lowercase f x. So we basically want to find capital F of X in this problem, and we are given lowercase F of X. Which is equal to -5 halves cosequant of 5 halves x. Cotingents of 5 halves X. What we have to remember is that the derivative of cosequant of AX according to the chain rule is equal to negative a multiplied by cosequant of AX multiplied by cotangent of AX, right? And specifically, if we understand this derivative using the tables of derivatives, we can solve this problem. What we're going to do is simply take our function and equate it to the right hand side of this relationship. In particular, we can show that negative. For now, we can ignore the constant in front, it's -5 halves. Let's just ignore it for now and let's suppose that cosequence. Of AX multiplied by cotangent of AX. is equal to cosequence of 5 halves x cotangent of 5 halves X, right? From here by analogy, we can show that A is equal to 5 halves. So what we can do is now use the a value, right, and show that the derivative of cosequant of 5 halves X is going to be equal to negative A, right? So -5 halves. Cosse Of 5 halves X. Cotingents of 5 halves X. Notice that we already have the function in question -5 halves go sequence of 5 halves x cotangent of 5 halves X. And in addition to that, we have to understand that this is only one of the NI derivatives. If we want to identify the whole family of functions F of X, we basically have to add a constant C because the derivative of a constant is equal to 0. So we can show that cosequant of 5 halves X plus C derivative is also equal to -5 halves, cosequant of 5 halves X, cotangent of 5/halves X. So now what is the answer? Well, on the left hand side we have the derivative of the required function, which is basically the inside derivative of the right hand side, right? So the answer to this problem is co-sequence of 5 halves X plus C, which corresponds to the answer choice B. 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.−π csc (πx/2) cot (πx/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.
−π csc (πx/2) cot (πx/2)