Integral Exercise 13

We are asked to evaluate the following indefinite integral:

$$ \int \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1-x^2}} \, dx $$

To begin, we apply the linearity property of integration, which allows us to split the integral of a difference into the difference of two integrals:

$$ \int \frac{1}{\sqrt{x}} \, dx - \int \frac{1}{\sqrt{1 - x^2}} \, dx $$

We rewrite the first integrand using exponent notation:

$$ \int x^{-1/2} \, dx - \int \frac{1}{\sqrt{1 - x^2}} \, dx $$

At this point, both integrals are elementary and can be evaluated directly.

The first is a standard power rule integral:

$$ \left[ \frac{x^{1/2}}{1/2} \right] + c - \int \frac{1}{\sqrt{1 - x^2}} \, dx $$

$$ = 2\sqrt{x} + c - \int \frac{1}{\sqrt{1 - x^2}} \, dx $$

The second integral is a well-known antiderivative, corresponding to the arcsine function:

$$ 2\sqrt{x} + c - \arcsin x $$

Combining terms, the final result is:

$$ 2\sqrt{x} - \arcsin x + c $$

And that completes the solution.