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

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

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integral of sqrt(3x^2-2)

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

\frac{1}{2} x 3 x^{2} - 2 - \frac{1}{3} ln \frac{1}{2} (3 x^{2} - 2) + \frac{3}{2} x + C

Solution steps

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

Apply Integration By Parts

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

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

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

= x 3 x^{2} - 2 - (\frac{1}{2} x 3 x^{2} - 2 + \frac{1}{3} ln \frac{3 x ^{2} - 2}{2} + \frac{3}{2} x)

Simplify

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

= \frac{1}{2} x 3 x^{2} - 2 - \frac{1}{3} ln \frac{1}{2} (3 x^{2} - 2) + \frac{3}{2} x

Add a constant to the solution

= \frac{1}{2} x 3 x^{2} - 2 - \frac{1}{3} ln \frac{1}{2} (3 x^{2} - 2) + \frac{3}{2} x + C

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