Integration of inverse trigonometric functions

$$ 1. \ \int \sin^{-1}x \ dx = x \sin^{-1}x + \sqrt {1-x^2} +C $$

$$ 2. \ \int \cos^{-1}x \ dx = x \cos^{-1}x - \sqrt {1-x^2} +C $$

$$ 3. \ \int \tan^{-1}x \ dx = x \tan^{-1}x - \frac 12 \ln (x^2+1) +C $$

$$ 4. \ \int \cot^{-1}x \ dx = x \cot^{-1}x + \frac 12 \ln (x^2+1) +C $$

$$ 5. \ \int \frac {du}{ \sqrt {a^2-u^2}} = \sin^{-1} \frac ua +C $$

$$ 6. \ \int \frac {du}{ a^2+u^2} = \frac 1a \tan^{-1} \frac ua +C $$

$$ 7. \ \int \frac {du}{u \sqrt {u^2-a^2}} = \frac 1a \sec^{-1} \frac ua +C $$

Example 1:

$$ Evaluate \ the \ Integral \int \frac {dx}{ \sqrt {4-9x^2}} $$

Solution:

$$ Substitute \ u=3x. \ Then \ du=3dx $$

$$ \int \frac {dx}{ \sqrt {4-9x^2}} = \frac 13 \frac {du}{ \sqrt {4-u^2}} $$

$$ = \frac 13 \sin^{-1} \frac u2 +C $$

$$ = \frac 13 \sin^{-1} \frac {3x}{2} +C $$

Example 2:

$$ Evaluate \ the \ Integral \int \frac {dx}{9+x^2} $$

Solution:

$$ Apply \ the \ formula \ with \ a=3, $$

$$ \int \frac {dx}{9+x^2} = \frac 13 \tan^{-1} \frac x3 +C $$