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

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

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

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

\frac{x ^{2}}{2} + arctan (x) + C

Solution steps

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

Long division

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

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

\int x d x = \frac{x ^{2}}{2}

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

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

Add a constant to the solution

= \frac{x ^{2}}{2} + arctan (x) + C

(t) e^{4 t}

\int sin^{2} (2 x) d x

\int \frac{1}{9 x ^{2} - 49} d x

\frac{d}{d x} ((x^{2} - 1)^{3})

\frac{1}{t ^{2} - 1}

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