What is the integral of (5-x)/(sqrt(x^2-10x+3)) ?

The integral of (5-x)/(sqrt(x^2-10x+3)) is -sqrt(x^2-10x+3)+C

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integral of (5-x)/(sqrt(x^2-10x+3))

\int \frac{5 - x}{x ^{2} - 10 x + 3} d x

- x^{2} - 10 x + 3 + C

Solution steps

\int \frac{5 - x}{x ^{2} - 10 x + 3} d x

Expand

\frac{5 - x}{x ^{2} - 10 x + 3} : \frac{5}{x ^{2} - 10 x + 3} - \frac{x}{x ^{2} - 10 x + 3}

= \int \frac{5}{x ^{2} - 10 x + 3} - \frac{x}{x ^{2} - 10 x + 3} 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 ^{2} - 10 x + 3} d x - \int \frac{x}{x ^{2} - 10 x + 3} d x

\int \frac{5}{x ^{2} - 10 x + 3} d x = 5 ln \frac{1}{22} (x^{2} - 10 x + 3) + \frac{1}{22} (x - 5)

\int \frac{x}{x ^{2} - 10 x + 3} d x = x^{2} - 10 x + 3 + 5 ln \frac{1}{22} (x^{2} - 10 x + 3) + \frac{1}{22} (x - 5)

= 5 ln \frac{1}{22} (x^{2} - 10 x + 3) + \frac{1}{22} (x - 5) - (x^{2} - 10 x + 3 + 5 ln \frac{1}{22} (x^{2} - 10 x + 3) + \frac{1}{22} (x - 5))

Simplify

= - x^{2} - 10 x + 3

Add a constant to the solution

= - x^{2} - 10 x + 3 + C

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What is the integral of (5-x)/(sqrt(x^2-10x+3)) ?
The integral of (5-x)/(sqrt(x^2-10x+3)) is -sqrt(x^2-10x+3)+C