What is the integral of 1/(2sqrt(1-3v)) ?

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

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integral of 1/(2sqrt(1-3v))

\int \frac{1}{2 1 - 3 v} d v

- \frac{1}{3} 1 - 3 v + C

Solution steps

\int \frac{1}{2 1 - 3 v} d v

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{1 - 3 v} d v

Apply u-substitution

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

= \frac{1}{2} \cdot \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}{2} (- \frac{1}{3} \cdot \int \frac{1}{u} d u)

Apply the Power Rule

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

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

Substitute back

u = 1 - 3 v

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

Simplify

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

= - \frac{1}{3} 1 - 3 v

Add a constant to the solution

= - \frac{1}{3} 1 - 3 v + C

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