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

The integral of (x^2-15L)/(x+7) is ((7+x)^2)/2-14(7+x)+49ln|7+x|-15Lln|7+x|+C

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

What is the integral of (x^2-15L)/(x+7) ?
The integral of (x^2-15L)/(x+7) is ((7+x)^2)/2-14(7+x)+49ln|7+x|-15Lln|7+x|+C