# How to Integrate Arcsin(x) or Sin-1(x)

Graphically, the inverse sine function looks like a small segment of the sine function reflected over the diagonal line y = x. In trigonometric terms, the function y = arcsin(x) or y = sin^{-1}(x) means that if you input a number x between -1 and 1, the function will out put an angle measure y between 0 and π radians such that the sine of y equals x. In other words, the function y = arcsin(x) is equivalent to sin(y) = x.

To find the antiderivative of f(x) = arcsin(x) you must use a combination of integration by parts and u-substitution. Here the integral of arsin(x) is worked out step by step.

## Graph of y = arcsin(x)

## Integral of y = arcsin(x)

The first step to integrate the function f(x) = arcsin(x) is to apply the standard trick of integration by parts. The integration by parts formula is

∫ u * dv = u * v - ∫ v * du

In the integral ∫ arcsin(x) dx, we let u = arcsin(x), dx = dv, du = 1/sqrt(1 - x^2) dx, and v = x. This gives us the integral equation

∫ arcsin(x) dx = x*arcsin(x) - ∫ x/sqrt(1 - x^2) dx

The right-hand side of this equation contains a new integral that looks slightly complicated, but we can still work it out if we apply the trick of substitution. Let's set w = x^2 and dw = 2x dx. Now we get the integral equivalence

∫ x/sqrt(1 - x^2) dx

= (1/2) * ∫ 1/sqrt(1 - w) dw

= -sqrt(1 - w) + c

= -sqrt(1 - x^2) + c

If we put all the separate pieces together, we get the final expression for the antiderivative of the function y = arcsin(x).

∫ arcsin(x) dx

= x*arcsin(x) + sqrt(1 - x^2) + c

## How to Use the Integral of arcsin(x)

The integral of the function y = arcsin(x) can be used to find the area under a curve, or to solve certain types of differential equations. The same steps above can also be used to integrate y = arccos(x), the inverse cosine, a closely related function. The integral of arccos(x) works out to be

∫ arccos(x) dx = x*arccos(x) - sqrt(1 - x^2) + c

Here are some more examples of how to use the integral of arcsin(x).

## Example 1

What is the area of the region bounded by the y-axis, the line y = 0.5, and the curve y = sin(x)? The region is shaded in yellow in the graph below.

As a simple integral, the area of the region is best represented by

∫ arcsin(x) dx, on the interval 0 ≤ x ≤ 0.7

The area is then found by plugging the endpoints of the interval into the antederivative and subtracting. This gives us

[0.7*arcsin(0.7) + sqrt(1 - 0.49)] - [0*arcsin(0) + sqrt(1 - 0)]

= 0.256921

## Example 2

Antiderivatives of functions are useful in find the general solutions to certain classes of differential equations. For example, the differential equation y' = x/arcsin(y). To solve this wieth separation of variables, we rewrite it as

dy / dx = x / arcsin(y)

and then separate the x and y expressions.

arcsin(y) dy = x dx

∫ arcsin(y) dy = ∫ x dx

y*arcsin(y) + sqrt(1 - y^2) = (1/2)x^2 + c

This equation cannot be solved for y in terms of x, so this expression is as simplified as it gets.

## Arcsin in Algebra Problems

Inverse trig functions are used to solve problems in regular algebra or pre-calculus. For example, suppose you want to find all the values of x such that

5*sin(πx) + 7 = 2π

Solving this equation for x gives us

5*sin(πx) = 2π - 7

sin(πx) = (2π - 7)/5

πx = arcsin[(2π - 7)/5]

πx = -0.1438586

x = -0.1438586π

x = -0.0457916

This is just one value of x that solves the equation. Since the function sin(πx) is periodic with a period of 2 and is symmetric about the line x = 1/2, the other values of x that solve the equation can be found continuing the following patterns.

x = ...-3+0.0457916, -1+0.457916, 1+0.457916, 3+0.457916, 5+0.457916, ...

x = -4-0.457916, -2-0.457916, -0.45716, 2-0.457916, 4-0.457916, ...

## More by this Author

- 29
This article has been recently edited to reflect the most recent changes in both lotteries' structures. Powerball and Mega Millions are the two largest lotteries in the US in terms of the number of players, revenue...

- 27
Comprehensive illustrated list of names for both flat shapes and solid shapes. Geometry reference for elementary and middle school, as well as home school. Over 60 pictures of geometric shapes.

- 8
Two simple functions, sin(x)/x and cos(x)/x, can't be integrated the easy way. Here is how it's done.

## Comments 2 comments

Thank you for doing this. It's actually not that difficult or long, it just requires some thinking.