What is the integral of sqrt(a-bx) ?

The integral of sqrt(a-bx) is -(2(a-bx)^{3/2})/(3b)+C

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integral of sqrt(a-bx)

\int a - b x d x

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

Solution steps

\int a - b x d x

Apply u-substitution

: \int - \frac{u}{b} d u

= \int - \frac{u}{b} d u

Take the constant out:

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

= - \frac{1}{b} \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = a - b x

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of sqrt(a-bx) ?
The integral of sqrt(a-bx) is -(2(a-bx)^{3/2})/(3b)+C