How to Integrate √(1 – x²)

To integrate the function √(1 – x²), we can use a trigonometric substitution. The expression under the square root, 1 – x², suggests using the sine function, since we can set x = sin(θ). This will simplify the integration process.

Here’s how we can proceed:

1. First, make the substitution: x = sin(θ). This implies that dx = cos(θ) dθ.
2. Next, substitute x into the integral:

∫ √(1 - x²) dx = ∫ √(1 - sin²(θ)) cos(θ) dθ

3. Using the Pythagorean identity, we know that √(1 – sin²(θ)) = cos(θ). Thus, the integral becomes:

∫ cos²(θ) dθ

4. To integrate cos²(θ), we can use the identity cos²(θ) = (1 + cos(2θ))/2. Therefore, the integral now becomes:

∫ (1 + cos(2θ))/2 dθ = 1/2 ∫ 1 dθ + 1/2 ∫ cos(2θ) dθ

5. Integrating these separately gives:

1/2 θ + 1/4 sin(2θ) + C

6. Now, we need to revert back to our original variable x. Recall that x = sin(θ), which means θ = arcsin(x). Additionally, we use the double angle identity for sine: sin(2θ) = 2 sin(θ) cos(θ), where cos(θ) = √(1 – x²).
7. Substituting these back into our expression gives:

1/2 arcsin(x) + 1/4 (2x√(1 - x²)) + C

Putting it all together, the final result of the integration is:

1/2 arcsin(x) + 1/4 x√(1 - x²) + C

And that’s how you integrate √(1 – x²)!