What is the integral of (2x)/(x^2-10x+25) ?

The integral of (2x)/(x^2-10x+25) is 2(ln|x-5|-5/(x-5))+C

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integral of (2x)/(x^2-10x+25)

\int \frac{2 x}{x ^{2} - 10 x + 25} d x

2 (ln ∣ x - 5 ∣ - \frac{5}{x - 5}) + C

Solution steps

\int \frac{2 x}{x ^{2} - 10 x + 25} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{x ^{2} - 10 x + 25} d x

Complete the square

x^{2} - 10 x + 25 : (x - 5)^{2}

= 2 \cdot \int \frac{x}{( x - 5 ) ^{2}} d x

Apply u-substitution

= 2 \cdot \int \frac{u + 5}{u ^{2}} d u

Expand

\frac{u + 5}{u ^{2}} : \frac{1}{u} + \frac{5}{u ^{2}}

= 2 \cdot \int \frac{1}{u} + \frac{5}{u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{1}{u} d u + \int \frac{5}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{5}{u ^{2}} d u = - \frac{5}{u}

= 2 (ln ∣ u ∣ - \frac{5}{u})

Substitute back

u = x - 5

= 2 (ln ∣ x - 5 ∣ - \frac{5}{x - 5})

Add a constant to the solution

= 2 (ln ∣ x - 5 ∣ - \frac{5}{x - 5}) + C

What is the integral of (2x)/(x^2-10x+25) ?
The integral of (2x)/(x^2-10x+25) is 2(ln|x-5|-5/(x-5))+C