What is the integral of x/(sqrt(5-4x)) ?

The integral of x/(sqrt(5-4x)) is -1/12 (2x+5)sqrt(-4x+5)+C

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integral of x/(sqrt(5-4x))

\int \frac{x}{5 - 4 x} d x

- \frac{1}{12} (2 x + 5) - 4 x + 5 + C

Solution steps

\int \frac{x}{5 - 4 x} d x

Apply u-substitution

: \int \frac{u ^{2} - 5}{8} d u

= \int \frac{u ^{2} - 5}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int u^{2} - 5 d u

Apply the Sum Rule:

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

= \frac{1}{8} (\int u^{2} d u - \int 5 d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 5 d u = 5 u

= \frac{1}{8} (\frac{u ^{3}}{3} - 5 u)

Substitute back

u = 5 - 4 x

= \frac{1}{8} (\frac{( 5 - 4 x ) ^{3}}{3} - 5 5 - 4 x)

Simplify

\frac{1}{8} (\frac{( 5 - 4 x ) ^{3}}{3} - 5 5 - 4 x) : - \frac{1}{12} (2 x + 5) - 4 x + 5

= - \frac{1}{12} (2 x + 5) - 4 x + 5

Add a constant to the solution

= - \frac{1}{12} (2 x + 5) - 4 x + 5 + C

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What is the integral of x/(sqrt(5-4x)) ?
The integral of x/(sqrt(5-4x)) is -1/12 (2x+5)sqrt(-4x+5)+C