What is the integral of 1/(sqrt(9-81x^2)) ?

The integral of 1/(sqrt(9-81x^2)) is 1/9 arcsin(3x)+C

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integral of 1/(sqrt(9-81x^2))

\int \frac{1}{9 - 81 x ^{2}} d x

\frac{1}{9} arcsin (3 x) + C

Solution steps

\int \frac{1}{9 - 81 x ^{2}} d x

Apply Trigonometric Substitution

= \int \frac{1}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= \frac{1}{9} \cdot 1 \cdot u

Substitute back

u = arcsin (\frac{9}{3} x)

= \frac{1}{9} \cdot 1 \cdot arcsin (\frac{9}{3} x)

Simplify

\frac{1}{9} \cdot 1 \cdot arcsin (\frac{9}{3} x) : \frac{1}{9} arcsin (3 x)

= \frac{1}{9} arcsin (3 x)

Add a constant to the solution

= \frac{1}{9} arcsin (3 x) + C

What is the integral of 1/(sqrt(9-81x^2)) ?
The integral of 1/(sqrt(9-81x^2)) is 1/9 arcsin(3x)+C