What is the integral of-(x^2)/(2(5-x)) ?

The integral of-(x^2)/(2(5-x)) is -1/2 (-(x^2)/2-5x-25ln|-x+5|)+C

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integral of-(x^2)/(2(5-x))

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

- \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣) + C

Solution steps

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

Take the constant out:

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

= - \frac{1}{2} \cdot \int \frac{x ^{2}}{5 - x} d x

Long division

\frac{x ^{2}}{5 - x} : - x - 5 + \frac{25}{- x + 5}

= - \frac{1}{2} \cdot \int - x - 5 + \frac{25}{- x + 5} d x

Apply the Sum Rule:

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

= - \frac{1}{2} (- \int x d x - \int 5 d x + \int \frac{25}{- x + 5} d x)

\int x d x = \frac{x ^{2}}{2}

\int 5 d x = 5 x

\int \frac{25}{- x + 5} d x = - 25 ln ∣ - x + 5 ∣

= - \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣)

Add a constant to the solution

= - \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣) + C

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What is the integral of-(x^2)/(2(5-x)) ?
The integral of-(x^2)/(2(5-x)) is -1/2 (-(x^2)/2-5x-25ln|-x+5|)+C