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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

: \int \frac{1}{2} d u

= \int \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= \frac{1}{2} \cdot 1 \cdot u

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

y = \frac{7 x - 24}{x ^{2} + 2 x}

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\frac{d}{d x} (ln (\frac{e ^{x}}{x}))

\int \frac{1}{e ^{4 x}} d x

\frac{d}{d x} (5 x^{6})

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