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Let us find

in a couple of different ways.

**As an antiderivative**

We may know the derivative of arcsine. We can start with

This may be rewritten as us

Now we take let us differentiate this with respect to *x*, remembering that *y* is a function of *x*. This means that we need to apply the chain rule. We get

or

But

This we can see from looking at the unit circle below.

The arcsine of *x* gives us an angle, and the sine of that gives us the value along the horizontal axis – and in a unit circle, the vertical and horizontal values are connected through Pythagoras’s theorem to give the above result. This gives us that

and thus that

Below is a graph of the integrand vs. the integral.

**By the substitution method**

Now suppose we did not know this above derivative, then we could find the integral by doing the substitution

This is because we may recognize that we then can use the trigonometric identity

We get

as before.

**The above generalized a bit**

Now if we have

then we can pull out a factor of *a* from the square root to get

If we now let

we get

This we can get directly if we apply the rule

to get

** **

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