How to integrate arc sin x dx?

1 Answer
Mar 4, 2016

intarcsin(x)dx = xarcsin(x)+sqrt(1-x^2)+C

Explanation:

We will proceed by using integration by substitution and integration by parts.

Substitution:

Let t = arcsin(x) => x = sin(t) and dx = cos(t)dt

Then, substituting, we have

intarcsin(x)dx = inttcos(t)dt

Integration by Parts:

Let u = t and dv = cos(t)dt

Then du = dt and v = sin(t)

By the integration by parts formula intudv = uv - intvdu

inttcos(t)dt = tsin(t)-intsint(t)dt

=tsint(t)-(-cos(t)+C)

=tsin(t)+cos(t)+C

=arcsin(x)*sin(arcsin(x))+cos(arcsin(x))+C

As sin(arcsin(x)) = x and cos(arcsin(x)) = sqrt(1-x^2)

(try drawing a right triangle where sin(theta)=x and calculate cos(theta) to obtain the second equality)

we obtain our final result:

intarcsin(x)dx = xarcsin(x)+sqrt(1-x^2)+C