What is the integral of (3x+3)/3 ?

The integral of (3x+3)/3 is 1/2 x(x+2)+C

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integral of (3x+3)/3

\int \frac{3 x + 3}{3} d x

\frac{1}{2} x (x + 2) + C

Solution steps

\int \frac{3 x + 3}{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 3 x + 3 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 3 x d x + \int 3 d x)

\int 3 x d x = \frac{3 x ^{2}}{2}

\int 3 d x = 3 x

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

Simplify

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

= \frac{1}{2} x (x + 2)

Add a constant to the solution

= \frac{1}{2} x (x + 2) + C

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