What is the integral of (7-x)/((x-2)^2) ?

The integral of (7-x)/((x-2)^2) is -5/(x-2)-ln|x-2|+C

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integral of (7-x)/((x-2)^2)

\int \frac{7 - x}{( x - 2 ) ^{2}} d x

- \frac{5}{x - 2} - ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{7 - x}{( x - 2 ) ^{2}} d x

Expand

\frac{7 - x}{( x - 2 ) ^{2}} : \frac{7}{( x - 2 ) ^{2}} - \frac{x}{( x - 2 ) ^{2}}

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{7}{( x - 2 ) ^{2}} d x - \int \frac{x}{( x - 2 ) ^{2}} d x

\int \frac{7}{( x - 2 ) ^{2}} d x = - \frac{7}{x - 2}

\int \frac{x}{( x - 2 ) ^{2}} d x = ln ∣ x - 2 ∣ - \frac{2}{x - 2}

= - \frac{7}{x - 2} - (ln ∣ x - 2 ∣ - \frac{2}{x - 2})

Simplify

- \frac{7}{x - 2} - (ln ∣ x - 2 ∣ - \frac{2}{x - 2}) : - \frac{5}{x - 2} - ln ∣ x - 2 ∣

= - \frac{5}{x - 2} - ln ∣ x - 2 ∣

Add a constant to the solution

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

\int ln (x^{2} - x + 2) d x

x \to \frac{π}{2} lim (6 sec (x))

h \to 0 lim (m x + b)

\int e^{x^{2} + 2} d x

\frac{d}{d x} (- 8 cos (2 x))

What is the integral of (7-x)/((x-2)^2) ?
The integral of (7-x)/((x-2)^2) is -5/(x-2)-ln|x-2|+C