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

The integral of 1/(sqrt(1-3x)) is -2/3 sqrt(1-3x)+C

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

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

- \frac{2}{3} 1 - 3 x + C

Solution steps

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

Apply u-substitution

: \int - \frac{1}{3 u} d u

= \int - \frac{1}{3 u} 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 \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

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

Substitute back

u = 1 - 3 x

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

Simplify

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

= - \frac{2}{3} 1 - 3 x

Add a constant to the solution

= - \frac{2}{3} 1 - 3 x + C

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