What is the integral of (4x+5)/(sqrt(4x+6)) ?

The integral of (4x+5)/(sqrt(4x+6)) is 2/3 xsqrt(4x+6)+1/2 sqrt(4x+6)+C

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integral of (4x+5)/(sqrt(4x+6))

\int \frac{4 x + 5}{4 x + 6} d x

\frac{2}{3} x 4 x + 6 + \frac{1}{2} 4 x + 6 + C

Solution steps

\int \frac{4 x + 5}{4 x + 6} d x

Expand

\frac{4 x + 5}{4 x + 6} : \frac{4 x}{4 x + 6} + \frac{5}{4 x + 6}

= \int \frac{4 x}{4 x + 6} + \frac{5}{4 x + 6} 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{4 x}{4 x + 6} d x + \int \frac{5}{4 x + 6} d x

\int \frac{4 x}{4 x + 6} d x = \frac{2}{3} (x - 3) 4 x + 6

\int \frac{5}{4 x + 6} d x = \frac{5}{2} 4 x + 6

= \frac{2}{3} (x - 3) 4 x + 6 + \frac{5}{2} 4 x + 6

Simplify

\frac{2}{3} (x - 3) 4 x + 6 + \frac{5}{2} 4 x + 6 : \frac{2}{3} x 4 x + 6 + \frac{1}{2} 4 x + 6

= \frac{2}{3} x 4 x + 6 + \frac{1}{2} 4 x + 6

Add a constant to the solution

= \frac{2}{3} x 4 x + 6 + \frac{1}{2} 4 x + 6 + C

\frac{d}{d x} (\frac{e ^{- x}}{x ^{2}})

\int 8 t^{3} (1 + t^{4})^{10} d t

\int \frac{- 1}{x ( ln ( x ) ) ^{2}} d x

x^{2} + y^{2} = 17, (4, - 1)

What is the integral of (4x+5)/(sqrt(4x+6)) ?
The integral of (4x+5)/(sqrt(4x+6)) is 2/3 xsqrt(4x+6)+1/2 sqrt(4x+6)+C