What is the integral of (7x^3-3x^2+21x+6)/(x^3+3x) ?

The integral of (7x^3-3x^2+21x+6)/(x^3+3x) is 7x-5/2 ln|x^2+3|+2ln|x|+C

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integral of (7x^3-3x^2+21x+6)/(x^3+3x)

\int \frac{7 x ^{3} - 3 x ^{2} + 21 x + 6}{x ^{3} + 3 x} d x

7 x - \frac{5}{2} ln x^{2} + 3 + 2 ln ∣ x ∣ + C

Solution steps

\int \frac{7 x ^{3} - 3 x ^{2} + 21 x + 6}{x ^{3} + 3 x} d x

Expand

\frac{7 x ^{3} - 3 x ^{2} + 21 x + 6}{x ^{3} + 3 x} : \frac{7 x ^{3}}{x ^{3} + 3 x} - \frac{3 x ^{2}}{x ^{3} + 3 x} + \frac{21 x}{x ^{3} + 3 x} + \frac{6}{x ^{3} + 3 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{7 x ^{3}}{x ^{3} + 3 x} d x - \int \frac{3 x ^{2}}{x ^{3} + 3 x} d x + \int \frac{21 x}{x ^{3} + 3 x} d x + \int \frac{6}{x ^{3} + 3 x} d x

\int \frac{7 x ^{3}}{x ^{3} + 3 x} d x = - 73 arctan (\frac{x}{3}) + 7 x

\int \frac{3 x ^{2}}{x ^{3} + 3 x} d x = \frac{3}{2} ln x^{2} + 3

\int \frac{21 x}{x ^{3} + 3 x} d x = 73 arctan (\frac{x}{3})

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

= - 73 arctan (\frac{x}{3}) + 7 x - \frac{3}{2} ln x^{2} + 3 + 73 arctan (\frac{x}{3}) + 2 ln ∣ x ∣ - ln x^{2} + 3

Simplify

- 73 arctan (\frac{x}{3}) + 7 x - \frac{3}{2} ln x^{2} + 3 + 73 arctan (\frac{x}{3}) + 2 ln ∣ x ∣ - ln x^{2} + 3 : 7 x - \frac{5}{2} ln x^{2} + 3 + 2 ln ∣ x ∣

= 7 x - \frac{5}{2} ln x^{2} + 3 + 2 ln ∣ x ∣

Add a constant to the solution

= 7 x - \frac{5}{2} ln x^{2} + 3 + 2 ln ∣ x ∣ + C

\frac{d y}{d x} + 3 x^{2} y = sin (x) e^{- x^{3}}

\frac{d}{d x} (cos (sin (tan (2 π x))))

cos (x y) = 1 + sin (y)

\frac{d}{d x} (4 e^{3 x^{2}})

\frac{2 x ^{3} + 5}{4 x ^{2} + 7}

What is the integral of (7x^3-3x^2+21x+6)/(x^3+3x) ?
The integral of (7x^3-3x^2+21x+6)/(x^3+3x) is 7x-5/2 ln|x^2+3|+2ln|x|+C