# Integrating a trig function divided by a trig function

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In summary, the conversation discusses finding the arc length of a curve and solving a problem involving integration. The individual is unsure of how to solve the problem correctly and is considering using integration by parts.
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## Homework Statement

Find the arc length of the curve r=4/θ, for ∏/2 ≤ θ ≤ ∏

## Homework Equations

L= ∫ ds = ∫ √(r^2 + (dr/dθ)^2) dθ

## The Attempt at a Solution

After some calculations, and letting θ = tanx, I now have to find ∫ ((secx)^3/(tanx)^2). I am not sure how to do this, but i have found online that ∫ (secx)^3 = (1/2)sec(x)tan(x)+(1/2)ln|sec(x)+tan(x)|. Using integration by parts and letting u = 1/(tan(x))^2= (cot(x))^2 and du/dx = -2cot(x).(cosec(x))^2 is looking very tedious. How do I solve this problem correctly?

I don't see how you got $sec^3(x)$. I get just sec(x) in the numerator.

Eventually, I reduce it to
$$\int \frac{sin^3(x)}{cos^2(x)}dx$$

Factor out a sin(x) to use with the dx and convert the remaining $sin^2(x)$ to $1- cos^2(x)$.

## 1. What is the process for integrating a trig function divided by a trig function?

The process for integrating a trig function divided by a trig function involves using trigonometric identities and substitution. First, use a trigonometric identity to rewrite the expression as a single trig function. Then, use substitution to convert the remaining trig function into a variable u. Finally, use integration techniques such as u-substitution or integration by parts to find the final answer.

## 2. Can I use a calculator to integrate a trig function divided by a trig function?

Yes, many calculators have the ability to integrate functions, including trig functions. However, it is important to understand the steps involved in the integration process so that you can double check your calculator's answer and understand the solution.

## 3. How do I know which trigonometric identity to use when integrating?

Choosing the right trigonometric identity to use when integrating a trig function divided by a trig function can be tricky. It often involves trial and error and recognizing patterns in the expression. It is helpful to have a list of common trigonometric identities on hand to refer to.

## 4. Are there any special cases when integrating a trig function divided by a trig function?

Yes, there are some special cases when integrating a trig function divided by a trig function. For example, if the expression involves a sine and cosine function with equal coefficients, you can use the double angle identity to simplify the expression. In addition, if the expression is in the form of secant and tangent, you can use the substitution u = sec(x) + tan(x) to simplify the integration process.

## 5. Can I use integration by parts when integrating a trig function divided by a trig function?

Yes, integration by parts can be used when integrating a trig function divided by a trig function. However, it may not always be the most efficient method and it is important to consider other integration techniques such as u-substitution or trigonometric identities.

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