What is the integral of (5x+3)/(sqrt(7x-1)) ?

The integral of (5x+3)/(sqrt(7x-1)) is 10/147 (7x-1)^{3/2}+52/49 sqrt(7x-1)+C

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integral of (5x+3)/(sqrt(7x-1))

\int \frac{5 x + 3}{7 x - 1} d x

\frac{10}{147} (7 x - 1)^{\frac{3}{2}} + \frac{52}{49} 7 x - 1 + C

Solution steps

\int \frac{5 x + 3}{7 x - 1} d x

Expand

\frac{5 x + 3}{7 x - 1} : \frac{5 x}{7 x - 1} + \frac{3}{7 x - 1}

= \int \frac{5 x}{7 x - 1} + \frac{3}{7 x - 1} 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{5 x}{7 x - 1} d x + \int \frac{3}{7 x - 1} d x

\int \frac{5 x}{7 x - 1} d x = \frac{10}{49} (\frac{1}{3} (7 x - 1)^{\frac{3}{2}} + 7 x - 1)

\int \frac{3}{7 x - 1} d x = \frac{6}{7} 7 x - 1

= \frac{10}{49} (\frac{1}{3} (7 x - 1)^{\frac{3}{2}} + 7 x - 1) + \frac{6}{7} 7 x - 1

Simplify

\frac{10}{49} (\frac{1}{3} (7 x - 1)^{\frac{3}{2}} + 7 x - 1) + \frac{6}{7} 7 x - 1 : \frac{10}{147} (7 x - 1)^{\frac{3}{2}} + \frac{52}{49} 7 x - 1

= \frac{10}{147} (7 x - 1)^{\frac{3}{2}} + \frac{52}{49} 7 x - 1

Add a constant to the solution

= \frac{10}{147} (7 x - 1)^{\frac{3}{2}} + \frac{52}{49} 7 x - 1 + C

\frac{y}{5}

\frac{d ^{2}}{d x ^{2}} (12 x^{3} - 32 x^{2})

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

x^{5}

\frac{x}{1 + 2 x ^{2}}

What is the integral of (5x+3)/(sqrt(7x-1)) ?
The integral of (5x+3)/(sqrt(7x-1)) is 10/147 (7x-1)^{3/2}+52/49 sqrt(7x-1)+C