# Integral of inverse trig or inverse hyperbolic

• USN2ENG
In summary, the conversation discusses the solution to a given integral and confirms that the solution provided is correct. The individual also asks a follow-up question about using hyperbolic inverse functions in definite and indefinite integrals.

## Homework Statement

∫5/(4x√(9-16x2)dx

## Homework Equations

I am pretty sure this is in the form of ∫du/(u√(a2-u2)

## The Attempt at a Solution

setting u=4x a=3 and du=4dx so 1/4du=dx I get:

-5/12 sech-1(4x/3) + C

Is this right or am I using the wrong definition? Just trying to check my answers

Thanks for any help

Yes, that is correct.

Thanks, I really appreciate it!

One quick follow-up question to anyone who can help:

Some of the integral definitions involving hyperbolic inverse functions call for if a>u or u>a. I know that dealing with a definite integral we just use the limits of integration to figure that out, but what if we are dealing with an indefinite integral? How do you know then?

## 1. What is the formula for the integral of inverse trig functions?

The formula for the integral of inverse trig functions is ∫(dx)/(√(1-x²)) = sin⁻¹x + C.

## 2. What is the significance of the constant C in the integral of inverse trig functions?

The constant C represents the unknown constant of integration and is added to the result of the integral to account for all possible solutions.

## 3. Is the integral of inverse trig functions always defined?

No, the integral of inverse trig functions is only defined for certain values of x. For example, the integral of arctan(x) is only defined for values of x between -1 and 1.

## 4. How do you solve integrals of inverse hyperbolic functions?

To solve integrals of inverse hyperbolic functions, you can use the substitution method. This involves substituting the inverse hyperbolic function with its equivalent logarithmic expression and then using integration techniques to solve the resulting equation.

## 5. What is the relationship between inverse trig and inverse hyperbolic functions?

Inverse trig and inverse hyperbolic functions are related through the complex logarithm. Inverse trig functions are defined using the inverse of the circular functions, while inverse hyperbolic functions are defined using the inverse of the hyperbolic functions.

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