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

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

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

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

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

Solution steps

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

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

Simplify

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

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

Add a constant to the solution

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

\int - 75 x^{2} + 600 x + C d x

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

(x^{2} + y) d x + (x + e^{y}) d y = 0

\int_{0}^{l n (2)} \frac{2}{e ^{x} + 3} d x

2 t e^{- t}

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