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

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

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integral of (x^3)/(sqrt(x^2+2))

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

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

Solution steps

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

Apply u-substitution

: \int u^{2} - 2 d u

= \int u^{2} - 2 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

= \int u^{2} d u - \int 2 d u

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

\int 2 d u = 2 u

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

Substitute back

u = x^{2} + 2

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

Simplify

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

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

Add a constant to the solution

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

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