What is the integral of (9x^2+5)/(3x) ?

The integral of (9x^2+5)/(3x) is 1/3 ((9x^2)/2+5ln|x|)+C

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integral of (9x^2+5)/(3x)

\int \frac{9 x ^{2} + 5}{3 x} d x

\frac{1}{3} (\frac{9 x ^{2}}{2} + 5 ln ∣ x ∣) + C

Solution steps

\int \frac{9 x ^{2} + 5}{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{9 x ^{2} + 5}{x} d x

Expand

\frac{9 x ^{2} + 5}{x} : 9 x + \frac{5}{x}

= \frac{1}{3} \cdot \int 9 x + \frac{5}{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

= \frac{1}{3} (\int 9 x d x + \int \frac{5}{x} d x)

\int 9 x d x = \frac{9 x ^{2}}{2}

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{1}{3} (\frac{9 x ^{2}}{2} + 5 ln ∣ x ∣)

Add a constant to the solution

= \frac{1}{3} (\frac{9 x ^{2}}{2} + 5 ln ∣ x ∣) + C

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What is the integral of (9x^2+5)/(3x) ?
The integral of (9x^2+5)/(3x) is 1/3 ((9x^2)/2+5ln|x|)+C