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

The integral of x(2x+1)^3 is (8x^5)/5+3x^4+2x^3+(x^2)/2+C

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integral of x(2x+1)^3

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

\frac{8 x ^{5}}{5} + 3 x^{4} + 2 x^{3} + \frac{x ^{2}}{2} + C

Solution steps

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

Expand

x (2 x + 1)^{3} : 8 x^{4} + 12 x^{3} + 6 x^{2} + x

= \int 8 x^{4} + 12 x^{3} + 6 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 8 x^{4} d x + \int 12 x^{3} d x + \int 6 x^{2} d x + \int x d x

\int 8 x^{4} d x = \frac{8 x ^{5}}{5}

\int 12 x^{3} d x = 3 x^{4}

\int 6 x^{2} d x = 2 x^{3}

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

= \frac{8 x ^{5}}{5} + 3 x^{4} + 2 x^{3} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{8 x ^{5}}{5} + 3 x^{4} + 2 x^{3} + \frac{x ^{2}}{2} + C

What is the integral of x(2x+1)^3 ?
The integral of x(2x+1)^3 is (8x^5)/5+3x^4+2x^3+(x^2)/2+C