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

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

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

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

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

Solution steps

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

Apply Integration By Parts

: x^{3} - 2 - \int \frac{x ^{3} + 4}{2 x x ^{3} - 2} d x

= x^{3} - 2 - \int \frac{x ^{3} + 4}{2 x x ^{3} - 2} d x

\int \frac{x ^{3} + 4}{2 x x ^{3} - 2} d x = \frac{1}{3} (x^{3} - 2 + 22 arctan (\frac{x ^{3} - 2}{2}))

= x^{3} - 2 - \frac{1}{3} (x^{3} - 2 + 22 arctan (\frac{x ^{3} - 2}{2}))

Simplify

x^{3} - 2 - \frac{1}{3} (x^{3} - 2 + 22 arctan (\frac{x ^{3} - 2}{2})) : \frac{2}{3} x^{3} - 2 - \frac{2 2}{3} arctan (\frac{1}{2} (x^{3} - 2))

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

Add a constant to the solution

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

f (x) = \frac{1 - 2 t}{1 + 6 t}

f (x) = \frac{5 x}{7 x ^{2} + 1}

x \to - 2.4 lim ([x])

y = \frac{4}{1 + x ^{2}}, (1, 2)

\int \frac{6 - 2 x}{9 - x ^{2}} d x

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