What is the integral of (6x+14)/(sqrt(x-42)) ?

The integral of (6x+14)/(sqrt(x-42)) is 4(x-42)^{3/2}+532sqrt(x-42)+C

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integral of (6x+14)/(sqrt(x-42))

\int \frac{6 x + 14}{x - 42} d x

4 (x - 42)^{\frac{3}{2}} + 532 x - 42 + C

Solution steps

\int \frac{6 x + 14}{x - 42} d x

Expand

\frac{6 x + 14}{x - 42} : \frac{6 x}{x - 42} + \frac{14}{x - 42}

= \int \frac{6 x}{x - 42} + \frac{14}{x - 42} 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{6 x}{x - 42} d x + \int \frac{14}{x - 42} d x

\int \frac{6 x}{x - 42} d x = 12 (\frac{1}{3} (x - 42)^{\frac{3}{2}} + 42 x - 42)

\int \frac{14}{x - 42} d x = 28 x - 42

= 12 (\frac{1}{3} (x - 42)^{\frac{3}{2}} + 42 x - 42) + 28 x - 42

Simplify

12 (\frac{1}{3} (x - 42)^{\frac{3}{2}} + 42 x - 42) + 28 x - 42 : 4 (x - 42)^{\frac{3}{2}} + 532 x - 42

= 4 (x - 42)^{\frac{3}{2}} + 532 x - 42

Add a constant to the solution

= 4 (x - 42)^{\frac{3}{2}} + 532 x - 42 + C

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What is the integral of (6x+14)/(sqrt(x-42)) ?
The integral of (6x+14)/(sqrt(x-42)) is 4(x-42)^{3/2}+532sqrt(x-42)+C