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

The integral of (x^2+24x^3)/x is 8x^3+(x^2)/2+C

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

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

8 x^{3} + \frac{x ^{2}}{2} + C

Solution steps

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

Simplify

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

= \int x (24 x + 1) d x

Expand

x (24 x + 1) : 24 x^{2} + x

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

Apply the Sum Rule:

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

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

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

\int x d x = \frac{x ^{2}}{2}

= 8 x^{3} + \frac{x ^{2}}{2}

Add a constant to the solution

= 8 x^{3} + \frac{x ^{2}}{2} + C

\frac{d}{d x} (arcsinh (5 x^{3} - 4 x^{2} + 11))

f (x) = 2 x - 1 + \frac{6}{x}, at x = 2

\frac{\partial}{\partial x} (A (e^{2 x} + 2 e^{2 x} x))

(x^{2} + y^{2})^{3} = 8 x^{2} y^{2}, (1, 1)

\frac{\partial}{\partial x} (x^{2} + y^{2} - z^{2})

What is the integral of (x^2+24x^3)/x ?
The integral of (x^2+24x^3)/x is 8x^3+(x^2)/2+C