# Solve for the Anti-Derivative of cosx dx / sin^3x: Expert Answer and Explanation

• laker_gurl3
In summary, the conversation discusses taking the anti-derivative of cosx dx / sin^3x and confirms that the answer -1/2(sinx)^-2 + C is correct. The use of Maple to solve the problem is mentioned, as well as a tip to check the solution by taking the derivative. The conversation also playfully mentions being lazy.
laker_gurl3
was just wondering if this was the right answer..
take the anti-derivative of:

cosx dx / sin^3x

i got
-1/2(sinx)^-2 + C
is that right?

Thats what maple says, good job.

Just a tip, if your not sure, take the derivative and check if you get the original function.

Maple?? Maple?? ! Let u= sin x. Then du= cos du and the integrand becomes
$$\frac{cos x}{sin^3 x}dx= \frac{1}{u^3}du= u^{-3}du$$

The anti-derivative of that is $$\frac{1}{-3+1}u^{-3+1}+C= \frac{1}{2}u^{-2}+C= \frac{1}{2}\frac{1}{sin^2 t}+C$$

Good job, laker_gurl3

Nothing wrong with being lazy!

## 1. What are anti-derivatives and why are they important?

Anti-derivatives are the inverse operation of derivatives and are used to find the original function from its derivative. They are important because they allow us to solve problems involving rates of change and optimization.

## 2. How do you find anti-derivatives?

To find an anti-derivative, you must use the reverse of the rules for finding derivatives. This includes using the power rule, product rule, quotient rule, and chain rule in reverse. You can also use integration techniques such as substitution and integration by parts.

## 3. Can anti-derivatives help in solving real-world problems?

Yes, anti-derivatives can be used to solve various real-world problems, such as determining the velocity of a moving object or finding the area under a curve. They are also used in fields like physics, economics, and engineering to model and analyze real-world situations.

## 4. Are there any limitations to using anti-derivatives?

While anti-derivatives are a powerful tool in mathematics, there are some limitations to their use. They cannot be used to find exact solutions for some functions, and in some cases, the anti-derivative may not exist. Additionally, anti-derivatives can be challenging to find for complex functions and may require advanced techniques.

## 5. How can I check if my anti-derivative is correct?

You can check the correctness of your anti-derivative by taking the derivative of the function and seeing if it matches the original function. Another way is to use the Fundamental Theorem of Calculus, which states that the definite integral of a function is equal to the difference between its anti-derivative evaluated at the upper and lower limits of integration.

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