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

The integral of (sqrt(2))/x+x/4+2/(sqrt(x)) is sqrt(2)ln|x|+(x^2)/8+4sqrt(x)+C

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

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

2 ln ∣ x ∣ + \frac{x ^{2}}{8} + 4 x + C

Solution steps

\int \frac{2}{x} + \frac{x}{4} + \frac{2}{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

= \int \frac{2}{x} d x + \int \frac{x}{4} d x + \int \frac{2}{x} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{x}{4} d x = \frac{x ^{2}}{8}

\int \frac{2}{x} d x = 4 x

= 2 ln ∣ x ∣ + \frac{x ^{2}}{8} + 4 x

Add a constant to the solution

= 2 ln ∣ x ∣ + \frac{x ^{2}}{8} + 4 x + C

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What is the integral of (sqrt(2))/x+x/4+2/(sqrt(x)) ?
The integral of (sqrt(2))/x+x/4+2/(sqrt(x)) is sqrt(2)ln|x|+(x^2)/8+4sqrt(x)+C