What is the integral of (3x+4)/(sqrt(3x+5)) ?

The integral of (3x+4)/(sqrt(3x+5)) is 2/9 (3x+5)^{3/2}-2/3 sqrt(3x+5)+C

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integral of (3x+4)/(sqrt(3x+5))

\int \frac{3 x + 4}{3 x + 5} d x

\frac{2}{9} (3 x + 5)^{\frac{3}{2}} - \frac{2}{3} 3 x + 5 + C

Solution steps

\int \frac{3 x + 4}{3 x + 5} d x

Expand

\frac{3 x + 4}{3 x + 5} : \frac{3 x}{3 x + 5} + \frac{4}{3 x + 5}

= \int \frac{3 x}{3 x + 5} + \frac{4}{3 x + 5} d x

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{3 x}{3 x + 5} d x + \int \frac{4}{3 x + 5} d x

\int \frac{3 x}{3 x + 5} d x = \frac{2}{3} (\frac{1}{3} (3 x + 5)^{\frac{3}{2}} - 5 3 x + 5)

\int \frac{4}{3 x + 5} d x = \frac{8}{3} 3 x + 5

= \frac{2}{3} (\frac{1}{3} (3 x + 5)^{\frac{3}{2}} - 5 3 x + 5) + \frac{8}{3} 3 x + 5

Simplify

\frac{2}{3} (\frac{1}{3} (3 x + 5)^{\frac{3}{2}} - 5 3 x + 5) + \frac{8}{3} 3 x + 5 : \frac{2}{9} (3 x + 5)^{\frac{3}{2}} - \frac{2}{3} 3 x + 5

= \frac{2}{9} (3 x + 5)^{\frac{3}{2}} - \frac{2}{3} 3 x + 5

Add a constant to the solution

= \frac{2}{9} (3 x + 5)^{\frac{3}{2}} - \frac{2}{3} 3 x + 5 + C

\int_{a}^{p} \frac{b}{x ( 1 - x )} d x

n = 0 \sum \infty \frac{e ^{n}}{n}

\frac{d y}{d x} = 0.5 y + x

What is the integral of (3x+4)/(sqrt(3x+5)) ?
The integral of (3x+4)/(sqrt(3x+5)) is 2/9 (3x+5)^{3/2}-2/3 sqrt(3x+5)+C