How to integrate sqrt(1-x^2)?

To integrate sqrt(1-x^2), use the substitution x = sin(u).

To integrate sqrt(1-x^2), we can use the substitution x = sin(u). This substitution allows us to express the integrand in terms of trigonometric functions, which we can then integrate using standard techniques.

First, we need to find dx/dt in terms of du/dt. Using the chain rule, we have:

dx/dt = cos(u) du/dt

Solving for du/dt, we get:

du/dt = dx/dt / cos(u)

Substituting x = sin(u) and dx/dt = cos(u) du/dt, we get:

du/dt = cos(u) / cos(u) = 1

So, we can simplify the integral as follows:

∫sqrt(1-x^2) dx = ∫sqrt(1-sin^2(u)) cos(u) du

Using the identity cos^2(u) = 1 - sin^2(u), we can simplify the integrand further:

∫sqrt(1-x^2) dx = ∫cos^2(u) du

We can now integrate using the power-reducing formula for cosine:

∫cos^2(u) du = (1/2) ∫(1 + cos(2u)) du

= (1/2) (u + (1/2) sin(2u)) + C

Substituting back x = sin(u), we get:

∫sqrt(1-x^2) dx = (1/2) arcsin(x) + (1/4) sin(2arcsin(x)) + C

This is the final answer.