What is the integral of (2x+1)/(sqrt(x^2-4x+5)) ?

The integral of (2x+1)/(sqrt(x^2-4x+5)) is 2sqrt(x^2-4x+5)+5ln|sqrt(x^2-4x+5)+x-2|+C

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integral of (2x+1)/(sqrt(x^2-4x+5))

\int \frac{2 x + 1}{x ^{2} - 4 x + 5} d x

2 x^{2} - 4 x + 5 + 5 ln x^{2} - 4 x + 5 + x - 2 + C

Solution steps

\int \frac{2 x + 1}{x ^{2} - 4 x + 5} d x

Expand

\frac{2 x + 1}{x ^{2} - 4 x + 5} : \frac{2 x}{x ^{2} - 4 x + 5} + \frac{1}{x ^{2} - 4 x + 5}

= \int \frac{2 x}{x ^{2} - 4 x + 5} + \frac{1}{x ^{2} - 4 x + 5} 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{2 x}{x ^{2} - 4 x + 5} d x + \int \frac{1}{x ^{2} - 4 x + 5} d x

\int \frac{2 x}{x ^{2} - 4 x + 5} d x = 2 (x^{2} - 4 x + 5 + 2 ln x^{2} - 4 x + 5 + x - 2)

\int \frac{1}{x ^{2} - 4 x + 5} d x = ln x^{2} - 4 x + 5 + x - 2

= 2 (x^{2} - 4 x + 5 + 2 ln x^{2} - 4 x + 5 + x - 2) + ln x^{2} - 4 x + 5 + x - 2

Simplify

2 (x^{2} - 4 x + 5 + 2 ln x^{2} - 4 x + 5 + x - 2) + ln x^{2} - 4 x + 5 + x - 2 : 2 x^{2} - 4 x + 5 + 5 ln x^{2} - 4 x + 5 + x - 2

= 2 x^{2} - 4 x + 5 + 5 ln x^{2} - 4 x + 5 + x - 2

Add a constant to the solution

= 2 x^{2} - 4 x + 5 + 5 ln x^{2} - 4 x + 5 + x - 2 + C

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What is the integral of (2x+1)/(sqrt(x^2-4x+5)) ?
The integral of (2x+1)/(sqrt(x^2-4x+5)) is 2sqrt(x^2-4x+5)+5ln|sqrt(x^2-4x+5)+x-2|+C