What is the integral of (7x^3-x^2+8x+12)/(x^4+4x^2) ?

The integral of (7x^3-x^2+8x+12)/(x^4+4x^2) is 5/2 ln|x^2+4|-2arctan(x/2)+2ln(x)-3/x+C

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integral of (7x^3-x^2+8x+12)/(x^4+4x^2)

\int \frac{7 x ^{3} - x ^{2} + 8 x + 12}{x ^{4} + 4 x ^{2}} d x

\frac{5}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + 2 ln (x) - \frac{3}{x} + C

Solution steps

\int \frac{7 x ^{3} - x ^{2} + 8 x + 12}{x ^{4} + 4 x ^{2}} d x

Expand

\frac{7 x ^{3} - x ^{2} + 8 x + 12}{x ^{4} + 4 x ^{2}} : \frac{7 x ^{3}}{x ^{4} + 4 x ^{2}} - \frac{x ^{2}}{x ^{4} + 4 x ^{2}} + \frac{8 x}{x ^{4} + 4 x ^{2}} + \frac{12}{x ^{4} + 4 x ^{2}}

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 ^{4} + 4 x ^{2}} d x - \int \frac{x ^{2}}{x ^{4} + 4 x ^{2}} d x + \int \frac{8 x}{x ^{4} + 4 x ^{2}} d x + \int \frac{12}{x ^{4} + 4 x ^{2}} d x

\int \frac{7 x ^{3}}{x ^{4} + 4 x ^{2}} d x = \frac{7}{2} ln x^{2} + 4

\int \frac{x ^{2}}{x ^{4} + 4 x ^{2}} d x = \frac{1}{2} arctan (\frac{x}{2})

\int \frac{8 x}{x ^{4} + 4 x ^{2}} d x = 2 ln (x) - ln x^{2} + 4

\int \frac{12}{x ^{4} + 4 x ^{2}} d x = 12 (- \frac{1}{4 x} - \frac{1}{8} arctan (\frac{x}{2}))

= \frac{7}{2} ln x^{2} + 4 - \frac{1}{2} arctan (\frac{x}{2}) + 2 ln (x) - ln x^{2} + 4 + 12 (- \frac{1}{4 x} - \frac{1}{8} arctan (\frac{x}{2}))

Simplify

\frac{7}{2} ln x^{2} + 4 - \frac{1}{2} arctan (\frac{x}{2}) + 2 ln (x) - ln x^{2} + 4 + 12 (- \frac{1}{4 x} - \frac{1}{8} arctan (\frac{x}{2})) : \frac{5}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + 2 ln (x) - \frac{3}{x}

= \frac{5}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + 2 ln (x) - \frac{3}{x}

Add a constant to the solution

= \frac{5}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + 2 ln (x) - \frac{3}{x} + C

\frac{\partial}{\partial y} (x^{2} + \frac{y}{x})

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

x \to 0 - lim (\frac{x ^{2} - x}{x - 1})

\int_{0}^{2} 5 x^{9} d x

\int 2 e^{2 x} + 12 x^{3} + 4 x d x

What is the integral of (7x^3-x^2+8x+12)/(x^4+4x^2) ?
The integral of (7x^3-x^2+8x+12)/(x^4+4x^2) is 5/2 ln|x^2+4|-2arctan(x/2)+2ln(x)-3/x+C