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

The integral of (sqrt(x)+3/(sqrt(x)))^2 is (x^2)/2+6x+9ln|x|+C

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

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

\frac{x ^{2}}{2} + 6 x + 9 ln ∣ x ∣ + C

Solution steps

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

Expand

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

= \int x + 6 + \frac{9}{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 x d x + \int 6 d x + \int \frac{9}{x} d x

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

\int 6 d x = 6 x

\int \frac{9}{x} d x = 9 ln ∣ x ∣

= \frac{x ^{2}}{2} + 6 x + 9 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 6 x + 9 ln ∣ x ∣ + C

What is the integral of (sqrt(x)+3/(sqrt(x)))^2 ?
The integral of (sqrt(x)+3/(sqrt(x)))^2 is (x^2)/2+6x+9ln|x|+C