Math exercises integral 1

Solve this indefinite integral

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

Solved step by step

To solve the antiderivative, apply a property of the integral calculus. Transform the integral of a sum into a sum of integrals

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

I write the first term using power notation

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

Now the two integrals are immediate integrals.

The first integral is the integral of a power.

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

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

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

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

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

The second integral is the integral of the arcsine of x

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

The solution of the integral is

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