What is the integral of (2x^3+4x^2-5)/(x+3) ?

The integral of (2x^3+4x^2-5)/(x+3) is (2x^3)/3-x^2+6x-23ln|x+3|-54+C

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integral of (2x^3+4x^2-5)/(x+3)

\int \frac{2 x ^{3} + 4 x ^{2} - 5}{x + 3} d x

\frac{2 x ^{3}}{3} - x^{2} + 6 x - 23 ln ∣ x + 3 ∣ - 54 + C

Solution steps

\int \frac{2 x ^{3} + 4 x ^{2} - 5}{x + 3} d x

Expand

\frac{2 x ^{3} + 4 x ^{2} - 5}{x + 3} : \frac{2 x ^{3}}{x + 3} + \frac{4 x ^{2}}{x + 3} - \frac{5}{x + 3}

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{2 x ^{3}}{x + 3} d x + \int \frac{4 x ^{2}}{x + 3} d x - \int \frac{5}{x + 3} d x

\int \frac{2 x ^{3}}{x + 3} d x = \frac{2 x ^{3}}{3} - 3 x^{2} + 18 x - 54 ln ∣ x + 3 ∣

\int \frac{4 x ^{2}}{x + 3} d x = 4 (\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣)

\int \frac{5}{x + 3} d x = 5 ln ∣ x + 3 ∣

= \frac{2 x ^{3}}{3} - 3 x^{2} + 18 x - 54 ln ∣ x + 3 ∣ + 4 (\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣) - 5 ln ∣ x + 3 ∣

Simplify

\frac{2 x ^{3}}{3} - 3 x^{2} + 18 x - 54 ln ∣ x + 3 ∣ + 4 (\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣) - 5 ln ∣ x + 3 ∣ : \frac{2 x ^{3}}{3} - x^{2} + 6 x - 23 ln ∣ x + 3 ∣ - 54

= \frac{2 x ^{3}}{3} - x^{2} + 6 x - 23 ln ∣ x + 3 ∣ - 54

Add a constant to the solution

= \frac{2 x ^{3}}{3} - x^{2} + 6 x - 23 ln ∣ x + 3 ∣ - 54 + C

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

\frac{d}{d x} (3 x^{4} - 4 x^{3} - 12 x^{2} + 5)

\frac{d}{d x} (sin (\frac{π x}{9}))

\frac{\partial}{\partial x} (x y e^{- 8 y})

\frac{d}{d t} (4 cos (t))

What is the integral of (2x^3+4x^2-5)/(x+3) ?
The integral of (2x^3+4x^2-5)/(x+3) is (2x^3)/3-x^2+6x-23ln|x+3|-54+C