What is the integral of (x^3-125)/(x^2+5x+25) ?

The integral of (x^3-125)/(x^2+5x+25) is (x^2)/2-5x+C

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integral of (x^3-125)/(x^2+5x+25)

\int \frac{x ^{3} - 125}{x ^{2} + 5 x + 25} d x

\frac{x ^{2}}{2} - 5 x + C

Solution steps

\int \frac{x ^{3} - 125}{x ^{2} + 5 x + 25} d x

Long division

\frac{x ^{3} - 125}{x ^{2} + 5 x + 25} : x - 5

= \int 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

= \int x d x - \int 5 d x

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

\int 5 d x = 5 x

= \frac{x ^{2}}{2} - 5 x

Add a constant to the solution

= \frac{x ^{2}}{2} - 5 x + C

\int (x + 5)^{\frac{1}{2}} d x

\int (e^{- 0.2 x}) d x

\int \frac{5}{x ^{2} - 16} d x

\int 10 x (1 - x)^{5} d x

\int x^{2} (7 x^{3} - 3)^{5} d x

What is the integral of (x^3-125)/(x^2+5x+25) ?
The integral of (x^3-125)/(x^2+5x+25) is (x^2)/2-5x+C