What is the integral of (x^3+x+5)/(x^2+1) ?

The integral of (x^3+x+5)/(x^2+1) is (x^2)/2+5arctan(x)+C

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integral of (x^3+x+5)/(x^2+1)

\int \frac{x ^{3} + x + 5}{x ^{2} + 1} d x

\frac{x ^{2}}{2} + 5 arctan (x) + C

Solution steps

\int \frac{x ^{3} + x + 5}{x ^{2} + 1} d x

Long division

\frac{x ^{3} + x + 5}{x ^{2} + 1} : x + \frac{5}{x ^{2} + 1}

= \int x + \frac{5}{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{5}{x ^{2} + 1} d x

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

\int \frac{5}{x ^{2} + 1} d x = 5 arctan (x)

= \frac{x ^{2}}{2} + 5 arctan (x)

Add a constant to the solution

= \frac{x ^{2}}{2} + 5 arctan (x) + C

\frac{d N}{d t} = N (1 - 0.001 N)

\int \frac{( arctan ( x ) ) ^{8}}{x ^{2} + 1} d x

\int \frac{3}{665} x^{2} d x

\int \frac{1}{( 1 - y ) ( 1 + y )} d y

y^{^{''}} + 6 y^{^{'}} - 27 y = 0

What is the integral of (x^3+x+5)/(x^2+1) ?
The integral of (x^3+x+5)/(x^2+1) is (x^2)/2+5arctan(x)+C