What is the integral of sqrt(m+ny) ?

The integral of sqrt(m+ny) is (2(m+ny)^{3/2})/(3n)+C

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integral of sqrt(m+ny)

\int m + n y d y

\frac{2 ( m + n y ) ^{\frac{3}{2}}}{3 n} + C

Solution steps

\int m + n y d y

Apply u-substitution

= \int \frac{u}{n} d u

Take the constant out:

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

= \frac{1}{n} \cdot \int u d u

Apply the Power Rule

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

Substitute back

u = m + n y

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

Simplify

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

= \frac{2 ( m + n y ) ^{\frac{3}{2}}}{3 n}

Add a constant to the solution

= \frac{2 ( m + n y ) ^{\frac{3}{2}}}{3 n} + C

What is the integral of sqrt(m+ny) ?
The integral of sqrt(m+ny) is (2(m+ny)^{3/2})/(3n)+C