What is the integral from 0 to 4 of (1+2sqrt(x))^2 ?

The integral from 0 to 4 of (1+2sqrt(x))^2 is 172/3

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integral from 0 to 4 of (1+2sqrt(x))^2

\int_{0}^{4} (1 + 2 x)^{2} d x

\frac{172}{3}

Decimal

57.33333 \dots

Solution steps

\int_{0}^{4} (1 + 2 x)^{2} d x

Expand

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

= \int_{0}^{4} 1 + 4 x + 4 x 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_{0}^{4} 1 d x + \int_{0}^{4} 4 x d x + \int_{0}^{4} 4 x d x

\int_{0}^{4} 1 d x = 4

\int_{0}^{4} 4 x d x = \frac{64}{3}

\int_{0}^{4} 4 x d x = 32

= 4 + \frac{64}{3} + 32

Simplify

4 + \frac{64}{3} + 32 : \frac{172}{3}

= \frac{172}{3}

What is the integral from 0 to 4 of (1+2sqrt(x))^2 ?
The integral from 0 to 4 of (1+2sqrt(x))^2 is 172/3