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

The integral from-1 to 1 of x^2(x+1)^2 is 16/15

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integral from-1 to 1 of x^2(x+1)^2

\int_{- 1}^{1} x^{2} (x + 1)^{2} d x

\frac{16}{15}

Decimal

1.06666 \dots

Solution steps

\int_{- 1}^{1} x^{2} (x + 1)^{2} d x

Expand

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

= \int_{- 1}^{1} x^{4} + 2 x^{3} + 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

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

\int_{- 1}^{1} x^{4} d x = \frac{2}{5}

\int_{- 1}^{1} 2 x^{3} d x = 0

\int_{- 1}^{1} x^{2} d x = \frac{2}{3}

= \frac{2}{5} + 0 + \frac{2}{3}

Simplify

\frac{2}{5} + 0 + \frac{2}{3} : \frac{16}{15}

= \frac{16}{15}

What is the integral from-1 to 1 of x^2(x+1)^2 ?
The integral from-1 to 1 of x^2(x+1)^2 is 16/15