What is the integral of (5sqrt(1-3x))/4 ?

The integral of (5sqrt(1-3x))/4 is -5/18 (-3x+1)^{3/2}+C

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integral of (5sqrt(1-3x))/4

\int \frac{5 1 - 3 x}{4} d x

- \frac{5}{18} (- 3 x + 1)^{\frac{3}{2}} + C

Solution steps

\int \frac{5 1 - 3 x}{4} d x

Take the constant out:

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

= \frac{5}{4} \cdot \int 1 - 3 x d x

Apply u-substitution

: \int - \frac{u}{3} d u

= \frac{5}{4} \cdot \int - \frac{u}{3} d u

Take the constant out:

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

= \frac{5}{4} (- \frac{1}{3} \cdot \int u d u)

Apply the Power Rule

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

= \frac{5}{4} (- \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 1 - 3 x

= \frac{5}{4} (- \frac{1}{3} \cdot \frac{2}{3} (1 - 3 x)^{\frac{3}{2}})

Simplify

\frac{5}{4} (- \frac{1}{3} \cdot \frac{2}{3} (1 - 3 x)^{\frac{3}{2}}) : - \frac{5}{18} (- 3 x + 1)^{\frac{3}{2}}

= - \frac{5}{18} (- 3 x + 1)^{\frac{3}{2}}

Add a constant to the solution

= - \frac{5}{18} (- 3 x + 1)^{\frac{3}{2}} + C

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What is the integral of (5sqrt(1-3x))/4 ?
The integral of (5sqrt(1-3x))/4 is -5/18 (-3x+1)^{3/2}+C