What is the integral of x(3x^2+1) ?

The integral of x(3x^2+1) is (3x^4)/4+(x^2)/2+C

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integral of x(3x^2+1)

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

\frac{3 x ^{4}}{4} + \frac{x ^{2}}{2} + C

Solution steps

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

Expand

x (3 x^{2} + 1) : 3 x^{3} + x

= \int 3 x^{3} + 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 3 x^{3} d x + \int x d x

\int 3 x^{3} d x = \frac{3 x ^{4}}{4}

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

= \frac{3 x ^{4}}{4} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{3 x ^{4}}{4} + \frac{x ^{2}}{2} + C

\frac{d y}{d x} = \frac{e ^{x}}{y}

\int - π cos (π x) d x

\int \frac{1}{( 4 x ) ^{2} + 2 ^{2}} d x

\frac{d y}{d t} + \frac{2}{t} y = 4 t - 3 + \frac{2}{t}

x \to 10 lim (\frac{x - 1 - 3}{x - 10})

What is the integral of x(3x^2+1) ?
The integral of x(3x^2+1) is (3x^4)/4+(x^2)/2+C