What is the integral of (sqrt(x)+x^{1/3}+x)/(x^2) ?

The integral of (sqrt(x)+x^{1/3}+x)/(x^2) is -2/(sqrt(x))-3/(2x^{2/3)}+ln|x|+C

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integral of (sqrt(x)+x^{1/3}+x)/(x^2)

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

- \frac{2}{x} - \frac{3}{2 x ^{\frac{2}{3}}} + ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{x + x ^{\frac{1}{3}} + x}{x ^{2}} : \frac{1}{x ^{\frac{3}{2}}} + \frac{1}{x ^{\frac{5}{3}}} + \frac{1}{x}

= \int \frac{1}{x ^{\frac{3}{2}}} + \frac{1}{x ^{\frac{5}{3}}} + \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

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{2}{x ^{\frac{1}{2}}} - \frac{3}{2 x ^{\frac{2}{3}}} + ln ∣ x ∣

Apply radical rule:

= - \frac{2}{x} - \frac{3}{2 x ^{\frac{2}{3}}} + ln ∣ x ∣

Add a constant to the solution

= - \frac{2}{x} - \frac{3}{2 x ^{\frac{2}{3}}} + ln ∣ x ∣ + C

\int_{0}^{1} e^{- 2 x} d x

y = x, x = 4

\int tan^{7} (x) d x

y = (x^{5} + 1)^{6}

y = 4 x^{2} + 2

What is the integral of (sqrt(x)+x^{1/3}+x)/(x^2) ?
The integral of (sqrt(x)+x^{1/3}+x)/(x^2) is -2/(sqrt(x))-3/(2x^{2/3)}+ln|x|+C