What is the integral of 1/(sqrt(a-by)) ?

The integral of 1/(sqrt(a-by)) is -(2sqrt(a-by))/b+C

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integral of 1/(sqrt(a-by))

\int \frac{1}{a - b y} d y

- \frac{2 a - b y}{b} + C

Solution steps

\int \frac{1}{a - b y} d y

Apply u-substitution

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

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

Take the constant out:

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

= - \frac{1}{b} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

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

Substitute back

u = a - b y

= - \frac{1}{b} \cdot 2 (a - b y)^{\frac{1}{2}}

Simplify

- \frac{1}{b} \cdot 2 (a - b y)^{\frac{1}{2}} : - \frac{2 a - b y}{b}

= - \frac{2 a - b y}{b}

Add a constant to the solution

= - \frac{2 a - b y}{b} + C

y^{^{'}} = y (7 - y) - \frac{49}{4}, y (0) = a

y^{^{'}} = \frac{( 3 x ^{4} + y ^{4} )}{x y ^{3}}

\frac{d}{d x} (x^{3} - cos (x))

\int - 7 sin (x) d x

- \frac{7}{x ^{3}}

What is the integral of 1/(sqrt(a-by)) ?
The integral of 1/(sqrt(a-by)) is -(2sqrt(a-by))/b+C