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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{2}{3 ( u ^{2} + 1 )} d u

= \int \frac{2}{3 ( u ^{2} + 1 )} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= \frac{2}{3} arctan (u)

Substitute back

u = x^{3} - 1

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

Add a constant to the solution

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

\int \frac{- 3}{x ^{2}} + 11 - 4^{x} d x

sin^{5} (x^{3})

x \to 5 lim (∣ x - 5 ∣)

y^{^{'}} - \frac{y}{2} = e^{- t}

(e^{x} - y) d x + (e^{y} - x) d y = 0

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