What is the integral of (12x^2+x+2)/(x(x^2+1)) ?

The integral of (12x^2+x+2)/(x(x^2+1)) is 5ln|x^2+1|+arctan(x)+2ln|x|+C

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integral of (12x^2+x+2)/(x(x^2+1))

\int \frac{12 x ^{2} + x + 2}{x ( x ^{2} + 1 )} d x

5 ln x^{2} + 1 + arctan (x) + 2 ln ∣ x ∣ + C

Solution steps

\int \frac{12 x ^{2} + x + 2}{x ( x ^{2} + 1 )} d x

Expand

\frac{12 x ^{2} + x + 2}{x ( x ^{2} + 1 )} : \frac{12 x ^{2}}{x ( x ^{2} + 1 )} + \frac{x}{x ( x ^{2} + 1 )} + \frac{2}{x ( x ^{2} + 1 )}

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

\int \frac{12 x ^{2}}{x ( x ^{2} + 1 )} d x = 6 ln x^{2} + 1

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

\int \frac{2}{x ( x ^{2} + 1 )} d x = 2 ln ∣ x ∣ - ln x^{2} + 1

= 6 ln x^{2} + 1 + arctan (x) + 2 ln ∣ x ∣ - ln x^{2} + 1

Simplify

= 5 ln x^{2} + 1 + arctan (x) + 2 ln ∣ x ∣

Add a constant to the solution

= 5 ln x^{2} + 1 + arctan (x) + 2 ln ∣ x ∣ + C

n = 1 \sum \infty \frac{3}{n + 1}

f (x) = sin (8 x + 4)

\int \frac{48 x ^{2}}{( x - 21 ) ( x + 7 ) ^{2}} d x

\frac{d}{d x} (4 x + 4)

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

What is the integral of (12x^2+x+2)/(x(x^2+1)) ?
The integral of (12x^2+x+2)/(x(x^2+1)) is 5ln|x^2+1|+arctan(x)+2ln|x|+C