What is the integral of (sqrt(2x)+3x^2)/(4x) ?

The integral of (sqrt(2x)+3x^2)/(4x) is 1/4 (sqrt(2/x)x+sqrt(2)sqrt(x)+(3x^2)/2)+C

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integral of (sqrt(2x)+3x^2)/(4x)

\int \frac{2 x + 3 x ^{2}}{4 x} d x

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

Solution steps

\int \frac{2 x + 3 x ^{2}}{4 x} d x

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{2 x + 3 x ^{2}}{x} d x

Expand

\frac{2 x + 3 x ^{2}}{x} : \frac{2}{x} + 3 x

= \frac{1}{4} \cdot \int \frac{2}{x} + 3 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}{4} (\int \frac{2}{x} d x + \int 3 x d x)

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

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

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

Add a constant to the solution

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

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What is the integral of (sqrt(2x)+3x^2)/(4x) ?
The integral of (sqrt(2x)+3x^2)/(4x) is 1/4 (sqrt(2/x)x+sqrt(2)sqrt(x)+(3x^2)/2)+C