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

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

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integral of x(1+2x)^2

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Add a constant to the solution

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

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