What is the integral of (8x)/(sqrt(5x+6)) ?

The integral of (8x)/(sqrt(5x+6)) is 16/25 (1/3 (5x+6)^{3/2}-6sqrt(5x+6))+C

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integral of (8x)/(sqrt(5x+6))

\int \frac{8 x}{5 x + 6} d x

\frac{16}{25} (\frac{1}{3} (5 x + 6)^{\frac{3}{2}} - 6 5 x + 6) + C

Solution steps

\int \frac{8 x}{5 x + 6} d x

Take the constant out:

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

= 8 \cdot \int \frac{x}{5 x + 6} d x

Apply u-substitution

: \int \frac{2 ( u ^{2} - 6 )}{25} d u

= 8 \cdot \int \frac{2 ( u ^{2} - 6 )}{25} d u

Take the constant out:

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

= 8 \cdot \frac{2}{25} \cdot \int u^{2} - 6 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

= 8 \cdot \frac{2}{25} (\int u^{2} d u - \int 6 d u)

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

\int 6 d u = 6 u

= 8 \cdot \frac{2}{25} (\frac{u ^{3}}{3} - 6 u)

Substitute back

u = 5 x + 6

= 8 \cdot \frac{2}{25} (\frac{( 5 x + 6 ) ^{3}}{3} - 6 5 x + 6)

Simplify

8 \cdot \frac{2}{25} (\frac{( 5 x + 6 ) ^{3}}{3} - 6 5 x + 6) : \frac{16}{25} (\frac{1}{3} (5 x + 6)^{\frac{3}{2}} - 6 5 x + 6)

= \frac{16}{25} (\frac{1}{3} (5 x + 6)^{\frac{3}{2}} - 6 5 x + 6)

Add a constant to the solution

= \frac{16}{25} (\frac{1}{3} (5 x + 6)^{\frac{3}{2}} - 6 5 x + 6) + C

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What is the integral of (8x)/(sqrt(5x+6)) ?
The integral of (8x)/(sqrt(5x+6)) is 16/25 (1/3 (5x+6)^{3/2}-6sqrt(5x+6))+C