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

The integral of sqrt(1-3x) is -2/9 (1-3x)^{3/2}+C

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integral of sqrt(1-3x)

\int 1 - 3 x d x

- \frac{2}{9} (1 - 3 x)^{\frac{3}{2}} + C

Solution steps

\int 1 - 3 x d x

Apply u-substitution

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

= \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{1}{3} \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = 1 - 3 x

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

Simplify

- \frac{1}{3} \cdot \frac{2}{3} (1 - 3 x)^{\frac{3}{2}} : - \frac{2}{9} (1 - 3 x)^{\frac{3}{2}}

= - \frac{2}{9} (1 - 3 x)^{\frac{3}{2}}

Add a constant to the solution

= - \frac{2}{9} (1 - 3 x)^{\frac{3}{2}} + C

\int_{0}^{1} e^{- x^{2}} d x

x y \frac{d y}{d x} + y^{2} = 2 x

x \to + 0 lim (\frac{1}{x})

y^{^{'}} - \frac{7}{x} y = \frac{y ^{5}}{x ^{5}}

e^{5 (t - 2)}

What is the integral of sqrt(1-3x) ?
The integral of sqrt(1-3x) is -2/9 (1-3x)^{3/2}+C