Integral of
The integral of is .
How to work it out
- 1
We want an antiderivative of f(x) = 1 / sqrt(1 - ).
- 2
Method: standard integral: arcsin.
- 3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
- 4
Check by differentiating the answer; you must get the original function back:
- 5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 0.5:
LaTeX source of the formulas on this page
\[ \int \frac{1}{\sqrt{1 - x^{2}}}\,dx = \arcsin\left(x\right) + C \]
\[ \int \frac{1}{\sqrt{1 - x^{2}}}\,dx \]
\[ \frac{d}{dx}\left[\arcsin\left(x\right)\right] = \frac{1}{\sqrt{1 - x^{2}}} \]
\[ \int_{0}^{0.5} \frac{1}{\sqrt{1 - x^{2}}}\,dx = \Big[\arcsin\left(x\right)\Big]_{0}^{0.5} \approx 0.52359878 \]Definite integrals of the same function
| Interval | Value |
|---|---|
| from 0 to 0.5 | 0.52359878 |
| from 0 to 0.25 | 0.25268026 |
Questions
What is the integral of 1 / sqrt(1 - )?
The integral of is .
Why do we add + C?
Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.
What is the integral of 1 / sqrt(1 - ) from 0 to 0.5?
About 0.52359878.