What is the integral of y/(sqrt(2x+3y)) ?

The integral of y/(sqrt(2x+3y)) is ysqrt(2x+3y)+C

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integral of y/(sqrt(2x+3y))

\int \frac{y}{2 x + 3 y} d x

y 2 x + 3 y + C

Solution steps

\int \frac{y}{2 x + 3 y} d x

Take the constant out:

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

= y \cdot \int \frac{1}{2 x + 3 y} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= y \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= y \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= y \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 2 x + 3 y

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

Simplify

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

= y 2 x + 3 y

Add a constant to the solution

= y 2 x + 3 y + C

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