# What is the integral of sqrt(x^2-1)/x dx?

• funzone36
In summary, the conversation discusses the process of solving for the integral of sqrt(x^2-1)/x dx and the mistake that was made in the solution. The correct answer involves using the substitution x = csc(u) and results in the equation arcsec(x)=arctan(sqrt(x^2-1)). The conversation also mentions the importance of drawing diagrams when solving these types of problems.
funzone36

## Homework Statement

What is the integral of sqrt(x^2-1)/x dx?

## The Attempt at a Solution

∫ √(x^2 - 1) dx / x

let x = sec u: u = sec^-1(x) and tan u = √(x^2 - 1)

dx = sec u tan u du

now the integral becomes

∫ √sec^2(u) - 1) sec u tan u du / sec u

= ∫ tan u tan u du

= ∫ tan^2(u) du

= ∫ (sec^2(u) - 1) du

= ∫ (sec^2(u) du - ∫ du

= tan u - u + c

back substitute u = sec^-1(x) and tan u = √(x^2 - 1)

= √(x^2 - 1) - sec^-1(x) + c

However, the correct answer should be arccot(sqrt(x^2-1))+sqrt(x^2-1) + c. Can someone help me find what went wrong?

knownothing
Maple says that this integral is
sqrt(x^2-1)+arctan(1/(x^2-1))+c

Well, arctan(1/(x^2-1)) is the same as arccot(sqrt(x^2-1)). I just don't know the mistake I did when showing my steps.

funzone36, what you initially had was more or less correct. you can check by writing sec^-1(x) as arccos(1/x) and then differentiating.

The answer that you said was supposedly correct involving arccot is gotten by substituting x = csc(u) and turns out to be a slightly neater approach.

arcsec(x)=arctan(sqrt(x^2-1)). Draw a right triangle with hypotenuse x and leg 1. t=arcsec(x) is the angle between those two. What's tan(t)?

Thanks everywhere. My steps were actually correct. I learned that I should draw diagrams next time I do these problems.

## 1. What is the definition of an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is a fundamental tool in calculus and is used to solve various problems in mathematics and science.

## 2. What is the integral of a function?

The integral of a function is the sum of infinitely many rectangles with infinitely small widths that make up the area under the curve of the function. It is represented by the symbol ∫ and is used to find the total value of the function within a specific range.

## 3. How do you solve for the integral of sqrt(x^2-1)/x dx?

To solve for the integral of sqrt(x^2-1)/x dx, we first need to rewrite the expression as 1/sqrt(x^2-1) dx. Then, we can use a substitution method or integration by parts to solve the integral. After solving, we need to add a constant of integration to the final answer.

## 4. What is the significance of the integral of sqrt(x^2-1)/x dx?

The integral of sqrt(x^2-1)/x dx has various applications in physics and engineering. It is used to calculate the work done by a variable force, the center of mass of an object, and the arc length of a curve, among other things.

## 5. Are there any special techniques for solving this integral?

Yes, there are various techniques for solving integrals, including substitution, integration by parts, trigonometric substitution, and partial fractions. The choice of technique depends on the complexity of the integral and the type of function being integrated.

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