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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

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

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

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 u

Integral of a constant:

\int a d x = a x

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

Substitute back

u = arcsin (\frac{3}{2} x)

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

Simplify

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

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

Add a constant to the solution

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

x \to π - lim (cot (x))

8 e^{4 x}

x y - 5 y^{2} = 4

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\int 18 - 18 sin (x) d x

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