Evaluate the integral: I = integral dx/(sqrt(5 - x²))

Evaluate the integral: $$ I = \int \frac{dx}{\sqrt{5 - x^2}}. $$

Explanation of Method

The denominator is already in the standard form $\sqrt{a^2-x^2}$, so the well-known arcsine antiderivative formula applies immediately, with no algebraic setup required beyond identifying $a$.

Recognize the integrand as matching the standard $\int \frac{dx}{\sqrt{a^2-x^2}}$ form.
Identify $a$ from $a^2$.
Apply the standard arcsine formula directly.

Step 1 · Recognize the form of the integral

The given integral is of the form:

$$ \int \frac{dx}{\sqrt{a^2-x^2}}, $$

where $a=\sqrt5$. This is a standard integral, and the result is:

$$ \int \frac{dx}{\sqrt{a^2-x^2}} = \arcsin\left(\frac{x}{a}\right) + C. $$

Step 2 · Apply the formula

Using $a=\sqrt5$, we substitute into the formula:

$$ I = \arcsin\left(\frac{x}{\sqrt{5}}\right) + C. $$

Summary Table

Quantity	Value
$a$	$\sqrt5$
Antiderivative	$\arcsin(x/\sqrt5)$

Additional Notes

The domain of $I$ requires $|x| < \sqrt5$, so that $5-x^2 > 0$ and the square root is real and nonzero.
This is the simplest case of the arcsine family, with $b=1$ — compare Problems 47 and 48, which add a coefficient in front of $x^2$ and require the more general formula.
The result can be checked by differentiating $\arcsin(x/\sqrt5)$ with the chain rule and confirming it returns $\dfrac{1}{\sqrt{5-x^2}}$.

Result

The solution to the integral is:

$$ I = \arcsin\left(\frac{x}{\sqrt{5}}\right) + C. $$