What is the integral of 1/(sqrt(3^2-x^2)) ?

The integral of 1/(sqrt(3^2-x^2)) is arcsin(1/3 x)+C

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integral of 1/(sqrt(3^2-x^2))

\int \frac{1}{3 ^{2} - x ^{2}} d x

arcsin (\frac{1}{3} x) + C

Solution steps

\int \frac{1}{3 ^{2} - x ^{2}} d x

3^{2} - x^{2} = 3^{2} (1 - \frac{x ^{2}}{3 ^{2}})

= \int \frac{1}{3 ^{2} ( 1 - \frac{x ^{2}}{3 ^{2}} )} d x

Simplify

\frac{1}{3 ^{2} ( 1 - \frac{x ^{2}}{3 ^{2}} )} : \frac{1}{9 - x ^{2}}

= \int \frac{1}{9 - x ^{2}} d x

Apply Trigonometric Substitution

= \int 1 d u

Integral of a constant:

\int a d x = a x

= 1 \cdot u

Substitute back

u = arcsin (\frac{1}{3} x)

= 1 \cdot arcsin (\frac{1}{3} x)

Simplify

= arcsin (\frac{1}{3} x)

Add a constant to the solution

= arcsin (\frac{1}{3} x) + C

What is the integral of 1/(sqrt(3^2-x^2)) ?
The integral of 1/(sqrt(3^2-x^2)) is arcsin(1/3 x)+C