What is the integral of (x+2)/(sqrt(x^2-6x+10)) ?

The integral of (x+2)/(sqrt(x^2-6x+10)) is sqrt(x^2-6x+10)+5ln|sqrt(x^2-6x+10)+x-3|+C

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integral of (x+2)/(sqrt(x^2-6x+10))

\int \frac{x + 2}{x ^{2} - 6 x + 10} d x

x^{2} - 6 x + 10 + 5 ln x^{2} - 6 x + 10 + x - 3 + C

Solution steps

\int \frac{x + 2}{x ^{2} - 6 x + 10} d x

Expand

\frac{x + 2}{x ^{2} - 6 x + 10} : \frac{x}{x ^{2} - 6 x + 10} + \frac{2}{x ^{2} - 6 x + 10}

= \int \frac{x}{x ^{2} - 6 x + 10} + \frac{2}{x ^{2} - 6 x + 10} 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{x}{x ^{2} - 6 x + 10} d x + \int \frac{2}{x ^{2} - 6 x + 10} d x

\int \frac{x}{x ^{2} - 6 x + 10} d x = x^{2} - 6 x + 10 + 3 ln x^{2} - 6 x + 10 + x - 3

\int \frac{2}{x ^{2} - 6 x + 10} d x = 2 ln x^{2} - 6 x + 10 + x - 3

= x^{2} - 6 x + 10 + 3 ln x^{2} - 6 x + 10 + x - 3 + 2 ln x^{2} - 6 x + 10 + x - 3

Simplify

= x^{2} - 6 x + 10 + 5 ln x^{2} - 6 x + 10 + x - 3

Add a constant to the solution

= x^{2} - 6 x + 10 + 5 ln x^{2} - 6 x + 10 + x - 3 + C

\int \frac{6 x ^{2}}{5 4 x ^{3} - 3} d x

\int \frac{x - 5}{x ^{2} - x - 6} d x

\int \frac{3 x ^{2} - 5 x + 7}{x + 1} d x

\int (\frac{1}{3 x} - 3 x^{\frac{5}{2}}) d x

\int \frac{6 x ^{5} - x + 5 x ^{2}}{x ^{3}} d x

What is the integral of (x+2)/(sqrt(x^2-6x+10)) ?
The integral of (x+2)/(sqrt(x^2-6x+10)) is sqrt(x^2-6x+10)+5ln|sqrt(x^2-6x+10)+x-3|+C