What is the integral of (2x+8x^2) ?

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

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integral of (2x+8x^2)

\int (2 x + 8 x^{2}) d x

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

Solution steps

\int 2 x + 8 x^{2} 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 2 x d x + \int 8 x^{2} d x

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

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

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{y}{x ^{2}})

y = x^{2} - 4, y = 0

y^{^{''}} - 6 y^{^{'}} - 8 y = 0

x + \frac{1}{x}

\frac{\partial}{\partial x} (5 x^{4} + 2 x y^{2} + 1)