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

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

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

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

- a^{2} - r^{2} + C

Solution steps

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

Apply u-substitution

= \int - \frac{1}{2 u} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= - \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = a^{2} - r^{2}

= - \frac{1}{2} \cdot 2 (a^{2} - r^{2})^{\frac{1}{2}}

Simplify

- \frac{1}{2} \cdot 2 (a^{2} - r^{2})^{\frac{1}{2}} : - a^{2} - r^{2}

= - a^{2} - r^{2}

Add a constant to the solution

= - a^{2} - r^{2} + C

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