What is the integral from-2 to x of (x+2)/3 ?

The integral from-2 to x of (x+2)/3 is 1/6 (x^2+4x+4)

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integral from-2 to x of (x+2)/3

\int_{- 2}^{x} \frac{x + 2}{3} d x

\frac{1}{6} (x^{2} + 4 x + 4)

Solution steps

\int_{- 2}^{x} \frac{x + 2}{3} d x

Take the constant out:

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

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

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

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

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

= \frac{1}{3} (\frac{x ^{2}}{2} - 2 + 2 x + 4)

Simplify

\frac{1}{3} (\frac{x ^{2}}{2} - 2 + 2 x + 4) : \frac{1}{6} (x^{2} + 4 x + 4)

= \frac{1}{6} (x^{2} + 4 x + 4)

\frac{3}{x ^{9}}

x \to \infty lim ((x + 1)^{x + 2})

\frac{d x}{d t} = \frac{4 x ^{2} + 7 t 4 t ^{2} + x ^{2}}{4 t x}

\frac{d}{d x} (- \frac{81}{( 1 + 9 x ) ^{2}})

y = 5 x^{2}, y = 30 x - 45, [0, 45]

What is the integral from-2 to x of (x+2)/3 ?
The integral from-2 to x of (x+2)/3 is 1/6 (x^2+4x+4)