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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x^{2} (x + 2)^{2} d x

Expand

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

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

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

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

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

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

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

Add a constant to the solution

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

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