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

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

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integral of 1/(sqrt(1-81t^2))

\int \frac{1}{1 - 81 t ^{2}} d t

\frac{1}{9} arcsin (9 t) + C

Solution steps

\int \frac{1}{1 - 81 t ^{2}} d t

Apply Trigonometric Substitution

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

= \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 (9 t)

= \frac{1}{9} \cdot 1 \cdot arcsin (9 t)

Simplify

= \frac{1}{9} arcsin (9 t)

Add a constant to the solution

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

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

\int e^{6 - 5 x} d x

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

\int \frac{x}{1 - x} d x

\int 9 x^{2} + 16 d x

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