What is the integral of x^3sqrt(x^2+24) ?

The integral of x^3sqrt(x^2+24) is 1/5 (x^2+24)^{5/2}-8(x^2+24)^{3/2}+C

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integral of x^3sqrt(x^2+24)

\int x^{3} x^{2} + 24 d x

\frac{1}{5} (x^{2} + 24)^{\frac{5}{2}} - 8 (x^{2} + 24)^{\frac{3}{2}} + C

Solution steps

\int x^{3} x^{2} + 24 d x

Apply u-substitution

= \int (u^{2} - 24) u^{2} d u

Expand

(u^{2} - 24) u^{2} : u^{4} - 24 u^{2}

= \int u^{4} - 24 u^{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^{4} d u - \int 24 u^{2} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int 24 u^{2} d u = 8 u^{3}

= \frac{u ^{5}}{5} - 8 u^{3}

Substitute back

u = x^{2} + 24

= \frac{( x ^{2} + 24 ) ^{5}}{5} - 8 (x^{2} + 24)^{3}

Simplify

\frac{( x ^{2} + 24 ) ^{5}}{5} - 8 (x^{2} + 24)^{3} : \frac{1}{5} (x^{2} + 24)^{\frac{5}{2}} - 8 (x^{2} + 24)^{\frac{3}{2}}

= \frac{1}{5} (x^{2} + 24)^{\frac{5}{2}} - 8 (x^{2} + 24)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{5} (x^{2} + 24)^{\frac{5}{2}} - 8 (x^{2} + 24)^{\frac{3}{2}} + C

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

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(2 x y + 3 y^{2}) d x = (2 x y + x^{2}) d y

What is the integral of x^3sqrt(x^2+24) ?
The integral of x^3sqrt(x^2+24) is 1/5 (x^2+24)^{5/2}-8(x^2+24)^{3/2}+C