What is the integral of (1+x+x^2)/(x+x^3) ?

The integral of (1+x+x^2)/(x+x^3) is ln|x|+arctan(x)+C

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integral of (1+x+x^2)/(x+x^3)

\int \frac{1 + x + x ^{2}}{x + x ^{3}} d x

ln ∣ x ∣ + arctan (x) + C

Solution steps

\int \frac{1 + x + x ^{2}}{x + x ^{3}} d x

Expand

\frac{1 + x + x ^{2}}{x + x ^{3}} : \frac{1}{x + x ^{3}} + \frac{x}{x + x ^{3}} + \frac{x ^{2}}{x + 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{1}{x + x ^{3}} d x + \int \frac{x}{x + x ^{3}} d x + \int \frac{x ^{2}}{x + x ^{3}} d x

\int \frac{1}{x + x ^{3}} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

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

\int \frac{x ^{2}}{x + x ^{3}} d x = \frac{1}{2} ln x^{2} + 1

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

Simplify

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

= ln ∣ x ∣ + arctan (x)

Add a constant to the solution

= ln ∣ x ∣ + arctan (x) + C

What is the integral of (1+x+x^2)/(x+x^3) ?
The integral of (1+x+x^2)/(x+x^3) is ln|x|+arctan(x)+C