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

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

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

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

\frac{x ^{3}}{3} + x + arctan (x) + C

Solution steps

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

\int x^{2} d x = \frac{x ^{3}}{3}

\int 1 d x = x

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

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

Add a constant to the solution

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

\int \frac{5}{x ^{3}} d x

\int \frac{cos ( 3 x ) + 3}{sin ( 3 x )} d x

\frac{3}{s ( s + 3 ) + 6}

\frac{\partial}{\partial x} (∣ x ∣ y)

\frac{d}{d x} (\frac{e ^{3 x} + e ^{x^{2}}}{3})

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