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

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

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integral of (1-ax)^{1/2}

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

- \frac{2 ( 1 - a x ) ^{\frac{3}{2}}}{3 a} + C

Solution steps

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

Apply u-substitution

: \int - \frac{u ^{\frac{1}{2}}}{a} d u

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= - \frac{1}{a} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 - a x

= - \frac{1}{a} \cdot \frac{2}{3} (1 - a x)^{\frac{3}{2}}

Simplify

- \frac{1}{a} \cdot \frac{2}{3} (1 - a x)^{\frac{3}{2}} : - \frac{2 ( 1 - a x ) ^{\frac{3}{2}}}{3 a}

= - \frac{2 ( 1 - a x ) ^{\frac{3}{2}}}{3 a}

Add a constant to the solution

= - \frac{2 ( 1 - a x ) ^{\frac{3}{2}}}{3 a} + C

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