What is the integral of (12x^2+9x)/(3x) ?

The integral of (12x^2+9x)/(3x) is x(2x+3)+C

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integral of (12x^2+9x)/(3x)

\int \frac{12 x ^{2} + 9 x}{3 x} d x

x (2 x + 3) + C

Solution steps

\int \frac{12 x ^{2} + 9 x}{3 x} 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 \frac{12 x ^{2} + 9 x}{x} d x

Expand

\frac{12 x ^{2} + 9 x}{x} : \frac{12 x ^{2}}{x} + \frac{9 x}{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 \frac{12 x ^{2}}{x} d x + \int \frac{9 x}{x} d x)

\int \frac{12 x ^{2}}{x} d x = 6 x^{2}

\int \frac{9 x}{x} d x = 9 x

= \frac{1}{3} (6 x^{2} + 9 x)

Simplify

\frac{1}{3} (6 x^{2} + 9 x) : x (2 x + 3)

= x (2 x + 3)

Add a constant to the solution

= x (2 x + 3) + C

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What is the integral of (12x^2+9x)/(3x) ?
The integral of (12x^2+9x)/(3x) is x(2x+3)+C