Ellen's question at Yahoo Answers ( Int 4 (tan ^3 x) dx )

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In summary, the question is asking for the steps to solve the integral ∫ 4 (tan^3)x dx. The response provides a detailed explanation of the steps involved, including the use of substitution and integration by parts. The final solution is given as 2tan^2x + 4ln|cosx| + C. The responder also mentions that there may have been an issue with accessing the page on Yahoo Answers.
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Fernando Revilla
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Hello Ellen,

We have: $$\begin{aligned}\int 4\tan^3 x\;dx&=4\int (\tan^2x)(\tan x)\:dx\\&=4\int(\sec^2x-1)(\tan x)dx\\&=4\int(\sec^2x)(\tan x)\:dx-4\int\tan x\;dx\end{aligned}$$ If $t=\tan x$, then $dt=\sec^2x\;dx$ so $$\int(\sec^2x)(\tan x)\:dx=\int t\:dt=\frac{t^2}{2}=\frac{\tan^2x}{2}$$ On the other hand:
$$\int\tan x\;dx=\int \frac{\sin x}{\cos x}dx=-\ln |\cos x|$$ As a consequence: $$\boxed{\;\displaystyle\int 4\tan^3 x\;dx=2\tan^2x+4\ln|\cos x|+C\;}$$

P.S. Something must be wrong with Yahoo Answers, when I'm logged in, I can't access to the corresponding page.

Edit: Now, that is all right.
 
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FAQ: Ellen's question at Yahoo Answers ( Int 4 (tan ^3 x) dx )

What is the meaning of "Int 4 (tan ^3 x) dx" in Ellen's question?

This is a notation used in calculus to represent the integral of the function (tan ^3 x) with respect to the variable x. The "Int" stands for integral and the "4" indicates that the integral is being taken exactly 4 times.

What is the purpose of taking the integral of (tan ^3 x) in this question?

The purpose is to find the area under the curve of the function (tan ^3 x) over a specific interval. In other words, it is used to calculate the total value of the function within a given range.

What is the process for solving this integral?

The process involves using techniques from calculus, such as integration by parts or substitution, to manipulate the given function and solve for the integral. It may also be helpful to use trigonometric identities to simplify the problem.

Why is it important to know how to solve this integral?

Integrals are widely used in mathematics, physics, and engineering to solve various problems. Knowing how to solve different types of integrals can help in solving real-world problems and understanding the behavior of functions.

Is there a general formula for solving integrals like this one?

There is no one general formula for solving all types of integrals. Different types of integrals require different techniques and methods for solving. However, there are some basic rules and formulas that can be applied in many cases.

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