What is the integral of (3x+7)/(sqrt(3x+8)) ?

The integral of (3x+7)/(sqrt(3x+8)) is 2/9 (3x+8)^{3/2}-2/3 sqrt(3x+8)+C

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integral of (3x+7)/(sqrt(3x+8))

\int \frac{3 x + 7}{3 x + 8} d x

\frac{2}{9} (3 x + 8)^{\frac{3}{2}} - \frac{2}{3} 3 x + 8 + C

Solution steps

\int \frac{3 x + 7}{3 x + 8} d x

Expand

\frac{3 x + 7}{3 x + 8} : \frac{3 x}{3 x + 8} + \frac{7}{3 x + 8}

= \int \frac{3 x}{3 x + 8} + \frac{7}{3 x + 8} 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 + 8} d x + \int \frac{7}{3 x + 8} d x

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

\int \frac{7}{3 x + 8} d x = \frac{14}{3} 3 x + 8

= \frac{2}{3} (\frac{1}{3} (3 x + 8)^{\frac{3}{2}} - 8 3 x + 8) + \frac{14}{3} 3 x + 8

Simplify

\frac{2}{3} (\frac{1}{3} (3 x + 8)^{\frac{3}{2}} - 8 3 x + 8) + \frac{14}{3} 3 x + 8 : \frac{2}{9} (3 x + 8)^{\frac{3}{2}} - \frac{2}{3} 3 x + 8

= \frac{2}{9} (3 x + 8)^{\frac{3}{2}} - \frac{2}{3} 3 x + 8

Add a constant to the solution

= \frac{2}{9} (3 x + 8)^{\frac{3}{2}} - \frac{2}{3} 3 x + 8 + C

y^{^{'}} = \frac{( 6 x ^{3} + y ^{3} )}{x y ^{2}}

y^{^{'}} = 1 - 100 (y - t)

\int_{0}^{\infty} \frac{1}{x ^{2}} d x

\int \frac{2 x + 1}{( x ^{2} + 6 x + 10 ) ( x + 1 )} d x

x \to - 6 lim (\frac{x ^{2} - 8}{6 - x})

What is the integral of (3x+7)/(sqrt(3x+8)) ?
The integral of (3x+7)/(sqrt(3x+8)) is 2/9 (3x+8)^{3/2}-2/3 sqrt(3x+8)+C