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

The integral from 1 to 3 of (x+1)^2 is 56/3

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integral from 1 to 3 of (x+1)^2

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

\frac{56}{3}

Decimal

18.66666 \dots

Solution steps

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

Expand

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

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

\int_{1}^{3} x^{2} d x = \frac{26}{3}

\int_{1}^{3} 2 x d x = 8

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

= \frac{26}{3} + 8 + 2

Simplify

\frac{26}{3} + 8 + 2 : \frac{56}{3}

= \frac{56}{3}

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What is the integral from 1 to 3 of (x+1)^2 ?
The integral from 1 to 3 of (x+1)^2 is 56/3