What is the integral of (2a)/(sqrt(2a+3v)) ?

The integral of (2a)/(sqrt(2a+3v)) is 4/3 asqrt(2a+3v)+C

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integral of (2a)/(sqrt(2a+3v))

\int \frac{2 a}{2 a + 3 v} d v

\frac{4}{3} a 2 a + 3 v + C

Solution steps

\int \frac{2 a}{2 a + 3 v} d v

Take the constant out:

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

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

Apply u-substitution

= 2 a \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

= 2 a \frac{1}{3} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

Substitute back

u = 2 a + 3 v

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

Simplify

2 a \frac{1}{3} \cdot 2 (2 a + 3 v)^{\frac{1}{2}} : \frac{4}{3} a 2 a + 3 v

= \frac{4}{3} a 2 a + 3 v

Add a constant to the solution

= \frac{4}{3} a 2 a + 3 v + C

What is the integral of (2a)/(sqrt(2a+3v)) ?
The integral of (2a)/(sqrt(2a+3v)) is 4/3 asqrt(2a+3v)+C