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

The integral of (3x+1)/(x^{2/3)} is 9/4 x^{4/3}+3x^{1/3}+C

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

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

\frac{9}{4} x^{\frac{4}{3}} + 3 x^{\frac{1}{3}} + C

Solution steps

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

Expand

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

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

\int 3 x^{\frac{1}{3}} d x = \frac{9}{4} x^{\frac{4}{3}}

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

= \frac{9}{4} x^{\frac{4}{3}} + 3 x^{\frac{1}{3}}

Add a constant to the solution

= \frac{9}{4} x^{\frac{4}{3}} + 3 x^{\frac{1}{3}} + C

\int \frac{1}{( cos ( x ) ) ^{3}} d x

y = 1 - x^{2}, y = 11 - 7 x

\int_{0}^{\frac{π}{2}} - 4 sin^{2} (3 x) d x

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

x \to 0 lim (\frac{tan ^{2} ( x )}{1 - cos ( x )})

What is the integral of (3x+1)/(x^{2/3)} ?
The integral of (3x+1)/(x^{2/3)} is 9/4 x^{4/3}+3x^{1/3}+C