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

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

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integral of sqrt(x)-1/2 x

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

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

Solution steps

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

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

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

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

Add a constant to the solution

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

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