What is the integral of r/(sqrt(r^2-x^2)) ?

The integral of r/(sqrt(r^2-x^2)) is rarcsin(x/r)+C

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integral of r/(sqrt(r^2-x^2))

\int \frac{r}{r ^{2} - x ^{2}} d x

r arcsin (\frac{x}{r}) + C

Solution steps

\int \frac{r}{r ^{2} - x ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= r \cdot \int \frac{1}{r ^{2} - x ^{2}} d x

Apply Trigonometric Substitution

= r \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= r \cdot 1 \cdot u

Substitute back

u = arcsin (\frac{1}{r} x)

= r \cdot 1 \cdot arcsin (\frac{1}{r} x)

Simplify

r \cdot 1 \cdot arcsin (\frac{1}{r} x) : r arcsin (\frac{x}{r})

= r arcsin (\frac{x}{r})

Add a constant to the solution

= r arcsin (\frac{x}{r}) + C

What is the integral of r/(sqrt(r^2-x^2)) ?
The integral of r/(sqrt(r^2-x^2)) is rarcsin(x/r)+C