What is the integral of ((3x^2-sqrt(x)))/(5x) ?

The integral of ((3x^2-sqrt(x)))/(5x) is 1/5 ((3x^2)/2-2sqrt(x))+C

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integral of ((3x^2-sqrt(x)))/(5x)

\int \frac{( 3 x ^{2} - x )}{5 x} d x

\frac{1}{5} (\frac{3 x ^{2}}{2} - 2 x) + C

Solution steps

\int \frac{3 x ^{2} - x}{5 x} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{3 x ^{2} - x}{x} d x

Expand

\frac{3 x ^{2} - x}{x} : 3 x - \frac{1}{x}

= \frac{1}{5} \cdot \int 3 x - \frac{1}{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}{5} (\int 3 x d x - \int \frac{1}{x} d x)

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

\int \frac{1}{x} d x = 2 x

= \frac{1}{5} (\frac{3 x ^{2}}{2} - 2 x)

Add a constant to the solution

= \frac{1}{5} (\frac{3 x ^{2}}{2} - 2 x) + C

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What is the integral of ((3x^2-sqrt(x)))/(5x) ?
The integral of ((3x^2-sqrt(x)))/(5x) is 1/5 ((3x^2)/2-2sqrt(x))+C