What is the integral of (x^3)/(\sqrt[9]{x^2+3)} ?

The integral of (x^3)/(\sqrt[9]{x^2+3)} is 1/2 (9/17 (x^2+3)^{17/9}-27/8 (x^2+3)^{8/9})+C

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integral of (x^3)/(\sqrt[9]{x^2+3)}

\frac{1}{2} (\frac{9}{17} (x^{2} + 3)^{\frac{17}{9}} - \frac{27}{8} (x^{2} + 3)^{\frac{8}{9}}) + C

Solution steps

Apply u-substitution

Take the constant out:

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

Expand

Apply the Sum Rule:

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

\int u^{\frac{8}{9}} d u = \frac{9}{17} u^{\frac{17}{9}}

= \frac{1}{2} (\frac{9}{17} u^{\frac{17}{9}} - \frac{27}{8} u^{\frac{8}{9}})

Substitute back

u = x^{2} + 3

= \frac{1}{2} (\frac{9}{17} (x^{2} + 3)^{\frac{17}{9}} - \frac{27}{8} (x^{2} + 3)^{\frac{8}{9}})

Add a constant to the solution

= \frac{1}{2} (\frac{9}{17} (x^{2} + 3)^{\frac{17}{9}} - \frac{27}{8} (x^{2} + 3)^{\frac{8}{9}}) + C

x \to 0 lim ((x + 5) e^{\frac{1}{x}})

\int_{0}^{π} (8 e^{x} + 7 sin (x)) d x

\frac{5 x ^{2} + 3 x}{2 x}

y = (x^{3} - 25 x)^{14}, (- 5, 0)

What is the integral of (x^3)/(\sqrt[9]{x^2+3)} ?
The integral of (x^3)/(\sqrt[9]{x^2+3)} is 1/2 (9/17 (x^2+3)^{17/9}-27/8 (x^2+3)^{8/9})+C