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

The integral of (2x^3+x+2)^2 is (4x^7)/7+(4x^5)/5+2x^4+(x^3)/3+2x^2+4x+C

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integral of (2x^3+x+2)^2

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

\frac{4 x ^{7}}{7} + \frac{4 x ^{5}}{5} + 2 x^{4} + \frac{x ^{3}}{3} + 2 x^{2} + 4 x + C

Solution steps

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

Expand

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

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

\int 4 x^{6} d x = \frac{4 x ^{7}}{7}

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

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

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

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

\int 4 d x = 4 x

= \frac{4 x ^{7}}{7} + \frac{4 x ^{5}}{5} + 2 x^{4} + \frac{x ^{3}}{3} + 2 x^{2} + 4 x

Add a constant to the solution

= \frac{4 x ^{7}}{7} + \frac{4 x ^{5}}{5} + 2 x^{4} + \frac{x ^{3}}{3} + 2 x^{2} + 4 x + C

What is the integral of (2x^3+x+2)^2 ?
The integral of (2x^3+x+2)^2 is (4x^7)/7+(4x^5)/5+2x^4+(x^3)/3+2x^2+4x+C