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

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

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

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

2 arctan (x) + x + C

Solution steps

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

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

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

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

\int 1 d x = x

= 2 arctan (x) + x

Add a constant to the solution

= 2 arctan (x) + x + C

x \to - 3 lim (3 x + 9)

\frac{3}{4} x^{\frac{4}{3}} - \frac{3}{8} x^{\frac{2}{3}} + 7

6 y^{^{''}} - y^{^{'}} - y = 0, y (0) = 10, y^{^{'}} (0) = 3

\frac{d}{d x} (\frac{x - x}{x + x})

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

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