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

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

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

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

\frac{3}{4} x^{\frac{4}{3}} - \frac{2}{3} x^{\frac{3}{2}} + x + C

Solution steps

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

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

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

\int 1 d x = x

= \frac{3}{4} x^{\frac{4}{3}} - \frac{2}{3} x^{\frac{3}{2}} + x

Add a constant to the solution

= \frac{3}{4} x^{\frac{4}{3}} - \frac{2}{3} x^{\frac{3}{2}} + x + C

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