∫1/(√1-x²)dx
To solve the integral ∫1/(√1-x²)dx, we can use a trigonometric substitution
To solve the integral ∫1/(√1-x²)dx, we can use a trigonometric substitution.
Let’s substitute x = sin(θ). Then, dx = cos(θ)dθ, and we need to find the integral in terms of θ.
Now, let’s rewrite the expression with the new substitution:
∫1/(√1-sin²(θ))cos(θ)dθ
Using the Pythagorean identity sin²(θ) + cos²(θ) = 1, we know that 1 – sin²(θ) = cos²(θ). Substituting this into the integral:
∫1/(√cos²(θ))cos(θ)dθ
Simplifying:
∫1/cos(θ)cos(θ)dθ
∫dθ
Now, the integral becomes simply θ + C, where C is the constant of integration.
However, we need to find the value of θ. To do that, we need to go back to our original substitution x = sin(θ).
From the original substitution, sin(θ) = x. Rearranging for θ, we have θ = arcsin(x).
Therefore, the final answer is θ + C, which in terms of x is arcsin(x) + C.
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