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

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

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

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

- \frac{2}{3} (3 - x)^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

= \int - u^{\frac{1}{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int u^{\frac{1}{2}} d u

Apply the Power Rule

= - \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 3 - x

= - \frac{2}{3} (3 - x)^{\frac{3}{2}}

Add a constant to the solution

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

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