# What is the Antiderivative of Inverse Sine of x?

• kristo
In summary, the conversation was about finding the antiderivative of \frac{1}{\sin\alpha}. The suggested method was to rewrite the sine in terms of exponential functions and use a change of variables and partial fractions to integrate it. Another suggestion was to rewrite it as {\csc\alpha} to make it more familiar. It was also mentioned that \frac{1}{\sin x} = \frac{\sin x}{1-\cos^2 x} is a useful formula to know.
kristo

## Homework Statement

Find the antiderivative of $$\frac{1}{\sin\alpha}$$

## The Attempt at a Solution

Ugh, maybe it's just a temporary brainfreeze, but I have no idea how to go about it.

Try rewriting the sine in terms of exponential functions. After a change of variables and partial fractions you should be able to integrate it.

kristo said:

## Homework Statement

Find the antiderivative of $$\frac{1}{\sin\alpha}$$

## The Attempt at a Solution

Ugh, maybe it's just a temporary brainfreeze, but I have no idea how to go about it.

Maybe if you write that as $${\csc\alpha}$$ it will look more familiar.

$$\frac{1}{\sin x} = \frac{\sin x}{1-\cos^2 x}$$

is a useful thing to know.

## 1. What is the antiderivative of sin(-1)x?

The antiderivative of sin(-1)x is cos(x) + C, where C is a constant.

## 2. How do you find the antiderivative of sin(-1)x?

To find the antiderivative of sin(-1)x, you can use the power rule for integration, which states that the antiderivative of x^n is (x^(n+1))/(n+1). In this case, n = -1, so the antiderivative is (x^-1)/-1 = -1/x. However, since we are dealing with sin(-1)x, we need to use the chain rule. This results in the antiderivative of sin(-1)x being cos(x) + C.

## 3. Can you show the steps for finding the antiderivative of sin(-1)x?

Step 1: Rewrite sin(-1)x as 1/sin(x) using the reciprocal identity for sine.Step 2: Use the power rule for integration, which gives us the antiderivative of 1/sin(x) as -1/x.Step 3: Use the chain rule by substituting -1/x back into the original expression sin(-1)x, resulting in cos(x) + C.

## 4. Is there a shortcut for finding the antiderivative of sin(-1)x?

Yes, there is a shortcut for finding the antiderivative of sin(-1)x. Since the derivative of cos(x) is -sin(x), it follows that the antiderivative of -sin(x) is cos(x). Therefore, the antiderivative of sin(-1)x is cos(x) + C.

## 5. What is the difference between the antiderivative of sin(-1)x and the integral of sin(-1)x?

The antiderivative of sin(-1)x is the function that, when differentiated, gives us sin(-1)x. The integral of sin(-1)x is the area under the curve of sin(-1)x between two points. In other words, the antiderivative is the reverse process of differentiation, while the integral is the reverse process of finding the area under a curve using integration.

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