What is the integral of (6x^2+x+7)/((x+1)(x^2+1)) ?

The integral of (6x^2+x+7)/((x+1)(x^2+1)) is 6ln|x+1|+arctan(x)+C

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integral of (6x^2+x+7)/((x+1)(x^2+1))

\int \frac{6 x ^{2} + x + 7}{( x + 1 ) ( x ^{2} + 1 )} d x

6 ln ∣ x + 1 ∣ + arctan (x) + C

Solution steps

\int \frac{6 x ^{2} + x + 7}{( x + 1 ) ( x ^{2} + 1 )} d x

Expand

\frac{6 x ^{2} + x + 7}{( x + 1 ) ( x ^{2} + 1 )} : \frac{6 x ^{2}}{( x + 1 ) ( x ^{2} + 1 )} + \frac{x}{( x + 1 ) ( x ^{2} + 1 )} + \frac{7}{( x + 1 ) ( 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{6 x ^{2}}{( x + 1 ) ( x ^{2} + 1 )} d x + \int \frac{x}{( x + 1 ) ( x ^{2} + 1 )} d x + \int \frac{7}{( x + 1 ) ( x ^{2} + 1 )} d x

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

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

\int \frac{7}{( x + 1 ) ( x ^{2} + 1 )} d x = 7 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x)))

= 6 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 - arctan (x))) - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 + arctan (x)) + 7 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x)))

Simplify

6 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 - arctan (x))) - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 + arctan (x)) + 7 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x))) : 6 ln ∣ x + 1 ∣ + arctan (x)

= 6 ln ∣ x + 1 ∣ + arctan (x)

Add a constant to the solution

= 6 ln ∣ x + 1 ∣ + arctan (x) + C

\frac{d y}{d x} = \frac{y}{x} + 4 x + 9

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

n = 0 \sum \infty \frac{1}{1.0 2 ^{n}}

\frac{d}{d T} (\frac{7}{6} T + 17)

\frac{d y}{d x} = \frac{y x + y}{y x + x}

What is the integral of (6x^2+x+7)/((x+1)(x^2+1)) ?
The integral of (6x^2+x+7)/((x+1)(x^2+1)) is 6ln|x+1|+arctan(x)+C