What is the integral of (x^2-15)/(x+7) ?

The integral of (x^2-15)/(x+7) is 1/2 (x+7)^2-14x-98+34ln|x+7|+C

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integral of (x^2-15)/(x+7)

\int \frac{x ^{2} - 15}{x + 7} d x

\frac{1}{2} (x + 7)^{2} - 14 x - 98 + 34 ln ∣ x + 7 ∣ + C

Solution steps

\int \frac{x ^{2} - 15}{x + 7} d x

Expand

\frac{x ^{2} - 15}{x + 7} : \frac{x ^{2}}{x + 7} - \frac{15}{x + 7}

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 ^{2}}{x + 7} d x - \int \frac{15}{x + 7} d x

\int \frac{x ^{2}}{x + 7} d x = \frac{( x + 7 ) ^{2}}{2} - 14 (x + 7) + 49 ln ∣ x + 7 ∣

\int \frac{15}{x + 7} d x = 15 ln ∣ x + 7 ∣

= \frac{( x + 7 ) ^{2}}{2} - 14 (x + 7) + 49 ln ∣ x + 7 ∣ - 15 ln ∣ x + 7 ∣

Simplify

\frac{( x + 7 ) ^{2}}{2} - 14 (x + 7) + 49 ln ∣ x + 7 ∣ - 15 ln ∣ x + 7 ∣ : \frac{1}{2} (x + 7)^{2} - 14 x - 98 + 34 ln ∣ x + 7 ∣

= \frac{1}{2} (x + 7)^{2} - 14 x - 98 + 34 ln ∣ x + 7 ∣

Add a constant to the solution

= \frac{1}{2} (x + 7)^{2} - 14 x - 98 + 34 ln ∣ x + 7 ∣ + C

What is the integral of (x^2-15)/(x+7) ?
The integral of (x^2-15)/(x+7) is 1/2 (x+7)^2-14x-98+34ln|x+7|+C