What is the integral of (x^2+1/(sqrt(x)))^2 ?

The integral of (x^2+1/(sqrt(x)))^2 is ln|x|+4/5 x^{5/2}+(x^5)/5+C

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integral of (x^2+1/(sqrt(x)))^2

\int (x^{2} + \frac{1}{x})^{2} d x

ln ∣ x ∣ + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{5}}{5} + C

Solution steps

\int (x^{2} + \frac{1}{x})^{2} d x

Expand

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

\int x^{4} d x = \frac{x ^{5}}{5}

= ln ∣ x ∣ + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{5}}{5}

Add a constant to the solution

= ln ∣ x ∣ + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{5}}{5} + C

- 384 e^{- 4 x}

\int \frac{3 x ^{2} - 10 x - 8}{x - 3} d x

\frac{d y}{d x} = - \frac{x}{y}

\int 5 tan^{3} (2 x) sec^{5} (2 x) d x

f (g (x))

What is the integral of (x^2+1/(sqrt(x)))^2 ?
The integral of (x^2+1/(sqrt(x)))^2 is ln|x|+4/5 x^{5/2}+(x^5)/5+C