What is the integral of 1/(sqrt(-3-12x-9x^2)) ?

The integral of 1/(sqrt(-3-12x-9x^2)) is 1/3 arcsin(3x+2)+C

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integral of 1/(sqrt(-3-12x-9x^2))

\int \frac{1}{- 3 - 12 x - 9 x ^{2}} d x

\frac{1}{3} arcsin (3 x + 2) + C

Solution steps

\int \frac{1}{- 3 - 12 x - 9 x ^{2}} d x

Complete the square

- 3 - 12 x - 9 x^{2} : - 9 (x + \frac{2}{3})^{2} + 1

= \int \frac{1}{- 9 ( x + \frac{2}{3} ) ^{2} + 1} d x

Apply u-substitution

: \int \frac{1}{- 9 u ^{2} + 1} d u

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

Apply Trigonometric Substitution

: \int \frac{1}{3} d v

= \int \frac{1}{3} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{3} \cdot 1 \cdot v

Substitute back

= \frac{1}{3} \cdot 1 \cdot arcsin (3 (x + \frac{2}{3}))

Simplify

\frac{1}{3} \cdot 1 \cdot arcsin (3 (x + \frac{2}{3})) : \frac{1}{3} arcsin (3 x + 2)

= \frac{1}{3} arcsin (3 x + 2)

Add a constant to the solution

= \frac{1}{3} arcsin (3 x + 2) + C

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What is the integral of 1/(sqrt(-3-12x-9x^2)) ?
The integral of 1/(sqrt(-3-12x-9x^2)) is 1/3 arcsin(3x+2)+C