What is the integral from 1 to 2 of a(a+2)^2 ?

The integral from 1 to 2 of a(a+2)^2 is 229/12

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integral from 1 to 2 of a(a+2)^2

\int_{1}^{2} a (a + 2)^{2} d a

\frac{229}{12}

Decimal Notation

19.08333 \dots

Solution steps

\int_{1}^{2} a (a + 2)^{2} d a

Expand

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

= \int_{1}^{2} a^{3} + 4 a^{2} + 4 a d a

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{2} a^{3} d a + \int_{1}^{2} 4 a^{2} d a + \int_{1}^{2} 4 a d a

\int_{1}^{2} a^{3} d a = \frac{15}{4}

\int_{1}^{2} 4 a^{2} d a = \frac{28}{3}

\int_{1}^{2} 4 a d a = 6

= \frac{15}{4} + \frac{28}{3} + 6

Simplify

\frac{15}{4} + \frac{28}{3} + 6 : \frac{229}{12}

= \frac{229}{12}

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What is the integral from 1 to 2 of a(a+2)^2 ?
The integral from 1 to 2 of a(a+2)^2 is 229/12