What is the integral of (5x-4)/(sqrt(8x-5x^2+9)) ?

The integral of (5x-4)/(sqrt(8x-5x^2+9)) is -sqrt(-5x^2+8x+9)+C

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integral of (5x-4)/(sqrt(8x-5x^2+9))

\int \frac{5 x - 4}{8 x - 5 x ^{2} + 9} d x

- - 5 x^{2} + 8 x + 9 + C

Solution steps

\int \frac{5 x - 4}{8 x - 5 x ^{2} + 9} d x

Apply u-substitution

: \int \frac{u}{5 - u ^{2} + 61} d u

= \int \frac{u}{5 - u ^{2} + 61} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{u}{- u ^{2} + 61} d u

Apply u-substitution

: \int - \frac{1}{2 v} d v

= \frac{1}{5} \cdot \int - \frac{1}{2 v} d v

Take the constant out:

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

= \frac{1}{5} (- \frac{1}{2} \cdot \int \frac{1}{v} d v)

Apply the Power Rule

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

= \frac{1}{5} (- \frac{1}{2} \cdot 2 v^{\frac{1}{2}})

Substitute back

= \frac{1}{5} (- \frac{1}{2} \cdot 2 (- (5 x - 4)^{2} + 61)^{\frac{1}{2}})

Simplify

\frac{1}{5} (- \frac{1}{2} \cdot 2 (- (5 x - 4)^{2} + 61)^{\frac{1}{2}}) : - - 5 x^{2} + 8 x + 9

= - - 5 x^{2} + 8 x + 9

Add a constant to the solution

= - - 5 x^{2} + 8 x + 9 + C

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What is the integral of (5x-4)/(sqrt(8x-5x^2+9)) ?
The integral of (5x-4)/(sqrt(8x-5x^2+9)) is -sqrt(-5x^2+8x+9)+C