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

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

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integral of x(2x+1)^2

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

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

Solution steps

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

Expand

x (2 x + 1)^{2} : 4 x^{3} + 4 x^{2} + x

= \int 4 x^{3} + 4 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 4 x^{3} d x + \int 4 x^{2} d x + \int x d x

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

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

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

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

Add a constant to the solution

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

f (x) = x^{2} + 4, (4, 20)

x \to 2 - lim (e^{\frac{1}{x - 2}})

y^{^{'}} = y (\frac{1}{x} - y)

y = x^{2} - 4 x - 5, y = 2 x - 5

\int_{2.25}^{2.75} 2 d x

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