What is the Antiderivative of Secant with a Non-Integer Power?

In summary, the conversation is about finding the integral of a function, specifically sec(t)^(8/3). The person tried to use a reduction formula, but realized it only works for integer powers. They then show how they simplified their initial problem and ask for help or clarification on whether they are on the right track. The other person responds that the integral probably does not have an elementary antiderivative and provides a more complex expression for it.
  • #1
Frillth
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0
For one of my homework assignments, I had to find the integral of a function. I got my function simplified to sec(t)^(8/3). I tried to use the reduction formula for sec(t)^n, but I believe that it only works if the power of sec is an integer. Could somebody help me out, please?

Edit: I figured that it might be a good idea if I showed how I got to sec(t)^(8/3)

My initial problem was the following: integral of cube root(1+x^2) dx.

First of all, I made the substitution x=tan(t) and dx = sec(t)^2 dt. This gave me:

integral of cube root(1+tan(t)^2) * sec(t)^2 dt.

I changed 1+tan(t)^2 to sec^2 to get the following:

integral of sec(t)^(2/3) * sec(t)^2 dt, or sec(t)^8/3.

Did I take this problem in the wrong direction, or am I on the right track?
 
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  • #2
It probably doesn't have an elementary antiderivative. Mathematica expresses it in terms of a hypergeometric function.
 

1. What is an antiderivative of secant?

An antiderivative of secant is a mathematical function that, when differentiated, gives the secant function. It is the inverse operation of differentiation and is denoted by the symbol ∫.

2. How do you find the antiderivative of secant?

To find the antiderivative of secant, you can use the formula ∫sec(x)dx = ln|sec(x) + tan(x)| + C, where C is a constant. This formula is derived from the chain rule of differentiation.

3. Can the antiderivative of secant be expressed in terms of elementary functions?

No, the antiderivative of secant cannot be expressed in terms of elementary functions, such as polynomials, trigonometric functions, and exponential functions. It is considered a non-elementary function.

4. What is the domain and range of the antiderivative of secant?

The domain of the antiderivative of secant is the set of all real numbers except for values where the secant function is undefined, such as at odd multiples of π/2. The range of the antiderivative of secant is also the set of all real numbers.

5. How is the antiderivative of secant used in real-world applications?

The antiderivative of secant has various applications in physics and engineering, particularly in the calculation of work done by a force. It is also used in the study of periodic functions and wave equations.

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