What is the integral of (x^5)/(sqrt(x^3+1)) ?

The integral of (x^5)/(sqrt(x^3+1)) is 2/3 (1/3 (x^3+1)^{3/2}-sqrt(x^3+1))+C

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integral of (x^5)/(sqrt(x^3+1))

\int \frac{x ^{5}}{x ^{3} + 1} d x

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

Solution steps

\int \frac{x ^{5}}{x ^{3} + 1} d x

Apply u-substitution

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

= \frac{2}{3} (\frac{u ^{3}}{3} - u)

Substitute back

u = x^{3} + 1

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (x^5)/(sqrt(x^3+1)) ?
The integral of (x^5)/(sqrt(x^3+1)) is 2/3 (1/3 (x^3+1)^{3/2}-sqrt(x^3+1))+C