integral of (2x-4)/((x^2-4x+20)^2)

1) Let's integrate:

\int \frac{2 x - 4}{{(x^{2} - 4 x + 20)}^{2}} d x

For the math expression:

\frac{2 x - 4}{{(x^{2} - 4 x + 20)}^{2}}

, since (a^n)' = n*a'*a^(n - 1), we try differentiating

{(x^{2} - 4 x + 20)}^{- 2 + 1}

2) Let's differentiate:

\frac{d}{d x} ({(x^{2} - 4 x + 20)}^{- 2 + 1})

ⓘ The exponent is computed before the power:

- 2 + 1

-1

\frac{d}{d x} ({(x^{2} - 4 x + 20)}^{-1})

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

(where f =

x^{2} - 4 x + 20

)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(x^{2} - 4 x + 20)}^{-1})

-1 \cdot \frac{d}{d x} (x^{2} - 4 x + 20) \cdot {(x^{2} - 4 x + 20)}^{-2}

-1 \cdot \frac{d}{d x} (x^{2} - 4 x + 20) \cdot {(x^{2} - 4 x + 20)}^{-2}

ⓘ Multiplying by -1 gives the opposite:

-1 \cdot \frac{d}{d x} (x^{2} - 4 x + 20) \cdot {(x^{2} - 4 x + 20)}^{-2}

- \frac{d}{d x} (x^{2} - 4 x + 20) \cdot {(x^{2} - 4 x + 20)}^{-2}

- \frac{d}{d x} (x^{2} - 4 x + 20) \cdot {(x^{2} - 4 x + 20)}^{-2}

ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

- \frac{d}{d x} (x^{2} - 4 x + 20) \cdot {(x^{2} - 4 x + 20)}^{-2}

- \frac{\frac{d}{d x} (x^{2} - 4 x + 20)}{{(x^{2} - 4 x + 20)}^{2}}

- \frac{\frac{d}{d x} (x^{2} - 4 x + 20)}{{(x^{2} - 4 x + 20)}^{2}}

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x^{2} - 4 x + 20)

\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 4 x) + 0

- \frac{\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 4 x) + 0}{{(x^{2} - 4 x + 20)}^{2}}

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

■Adding 0 changes nothing:

\frac{d}{d x} (- 4 x) + 0

\frac{d}{d x} (- 4 x)

■A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (- 4 x)

-4

- \frac{2 x - 4}{{(x^{2} - 4 x + 20)}^{2}}

ⓘ The minus sign of the fraction goes into the numerator or the denominator, changing the sign of each of its terms:

- \frac{2 x - 4}{{(x^{2} - 4 x + 20)}^{2}}

\frac{- 2 x + 4}{{(x^{2} - 4 x + 20)}^{2}}

\frac{- 2 x + 4}{{(x^{2} - 4 x + 20)}^{2}}

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

ⓘ The derivative of

- \frac{1}{x^{2} - 4 x + 20}

\frac{2 x - 4}{{(x^{2} - 4 x + 20)}^{2}}

, so

- \frac{1}{x^{2} - 4 x + 20}

is an antiderivative of

\frac{2 x - 4}{{(x^{2} - 4 x + 20)}^{2}}

\int \frac{2 x - 4}{{(x^{2} - 4 x + 20)}^{2}} d x

- \frac{1}{x^{2} - 4 x + 20}

- \frac{1}{x^{2} - 4 x + 20}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

- \frac{1}{x^{2} - 4 x + 20} + C