# Integrating arcsinx: Solving a Challenging Calculus Problem

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In summary, the formula for integrating arcsinx is ∫arcsinx dx = xarcsinx - √(x^2 - 1) + C. This formula can be proved using integration by parts and trigonometric identities, specifically the double angle formula for sine. The domain of arcsinx is [-1, 1] and the range is [-π/2, π/2]. Additionally, the integration of arcsinx can be solved using the substitution method by substituting u = sinx. There are several real-life applications of integrating arcsinx, including calculating surface area, finding arc length, and analyzing signals in various fields.

## Homework Statement

Intergrate arcsinx

## The Attempt at a Solution

I couldn't even touch this question. I saw the answer but still can not understand how to do this.

Use the "same" method as for arctan x and for ln x.

arcsin x = (arcsin x) * 1

Now integrate by parts!

Thanks a lot!

## 1. What is the formula for integrating arcsinx?

The formula for integrating arcsinx is ∫arcsinx dx = xarcsinx - √(x^2 - 1) + C.

## 2. How do you prove the integration formula for arcsinx?

The integration formula for arcsinx can be proved using integration by parts and trigonometric identities, specifically the double angle formula for sine.

## 3. What is the domain and range of arcsinx?

The domain of arcsinx is [-1, 1] and the range is [-π/2, π/2].

## 4. Can the integration of arcsinx be solved using substitution?

Yes, the integration of arcsinx can be solved using the substitution method, specifically by substituting u = sinx.

## 5. Are there any real-life applications of the integration of arcsinx?

Yes, the integration of arcsinx is used in various fields such as physics, engineering, and signal processing. It is used to calculate the surface area of a spherical cap, find the arc length of a circle segment, and analyze signals in communication systems.

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