What is the integral of ((y-3))/(sqrt(-24+14y-y^2)) ?

The integral of ((y-3))/(sqrt(-24+14y-y^2)) is -sqrt(-y^2+14y-24)+4arcsin(1/5 (y-7))+C

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integral of ((y-3))/(sqrt(-24+14y-y^2))

\int \frac{( y - 3 )}{- 24 + 14 y - y ^{2}} d y

- - y^{2} + 14 y - 24 + 4 arcsin (\frac{1}{5} (y - 7)) + C

Solution steps

\int \frac{y - 3}{- 24 + 14 y - y ^{2}} d y

Expand

\frac{y - 3}{- 24 + 14 y - y ^{2}} : \frac{y}{- 24 + 14 y - y ^{2}} - \frac{3}{- 24 + 14 y - y ^{2}}

= \int \frac{y}{- 24 + 14 y - y ^{2}} - \frac{3}{- 24 + 14 y - y ^{2}} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{y}{- 24 + 14 y - y ^{2}} d y - \int \frac{3}{- 24 + 14 y - y ^{2}} d y

\int \frac{y}{- 24 + 14 y - y ^{2}} d y = - - y^{2} + 14 y - 24 + 7 arcsin (\frac{1}{5} (y - 7))

\int \frac{3}{- 24 + 14 y - y ^{2}} d y = 3 arcsin (\frac{1}{5} (y - 7))

= - - y^{2} + 14 y - 24 + 7 arcsin (\frac{1}{5} (y - 7)) - 3 arcsin (\frac{1}{5} (y - 7))

Simplify

= - - y^{2} + 14 y - 24 + 4 arcsin (\frac{1}{5} (y - 7))

Add a constant to the solution

= - - y^{2} + 14 y - 24 + 4 arcsin (\frac{1}{5} (y - 7)) + C

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\int \frac{4}{1 - 16 x ^{2}} d x

What is the integral of ((y-3))/(sqrt(-24+14y-y^2)) ?
The integral of ((y-3))/(sqrt(-24+14y-y^2)) is -sqrt(-y^2+14y-24)+4arcsin(1/5 (y-7))+C