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

The integral of (x^3)/(sqrt(3x^2+\sqrt{2))} is

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

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

\frac{1}{18} (\frac{2}{3} (3 x^{2} + 2)^{\frac{3}{2}} - 3 \frac{2}{3 x ^{2} + 2} x^{2} - 2 \frac{2}{3 x ^{2} + 2} - 2 3 x^{2} + 2) + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 2}{18 u} d u

= \int \frac{u - 2}{18 u} d u

Take the constant out:

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

= \frac{1}{18} \cdot \int \frac{u - 2}{u} d u

Expand

\frac{u - 2}{u} : u - \frac{2}{u}

= \frac{1}{18} \cdot \int u - \frac{2}{u} 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{1}{18} (\int u d u - \int \frac{2}{u} d u)

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

\int \frac{2}{u} d u = \frac{2}{u} u + 2 u

= \frac{1}{18} (\frac{2}{3} u^{\frac{3}{2}} - (\frac{2}{u} u + 2 u))

Substitute back

u = 3 x^{2} + 2

= \frac{1}{18} (\frac{2}{3} (3 x^{2} + 2)^{\frac{3}{2}} - (\frac{2}{3 x ^{2} + 2} (3 x^{2} + 2) + 2 3 x^{2} + 2))

Expand

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

= \frac{1}{18} (\frac{2}{3} (3 x^{2} + 2)^{\frac{3}{2}} - 3 \frac{2}{3 x ^{2} + 2} x^{2} - 2 \frac{2}{3 x ^{2} + 2} - 2 3 x^{2} + 2)

Add a constant to the solution

= \frac{1}{18} (\frac{2}{3} (3 x^{2} + 2)^{\frac{3}{2}} - 3 \frac{2}{3 x ^{2} + 2} x^{2} - 2 \frac{2}{3 x ^{2} + 2} - 2 3 x^{2} + 2) + C

\frac{s}{s ( s + 1 )}

\int \frac{e ^{\frac{8}{x}}}{x ^{2}} d x

\frac{d}{d x} (- 2 sec^{2} (x) tan (x))

x \to 0 lim (- csc^{2} (x))

\frac{d}{d x} (3 a^{3 x})

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