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

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

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

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

ln x^{2} + 1 + 3 arctan (x) + C

Solution steps

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

Expand

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

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

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

\int \frac{3}{x ^{2} + 1} d x = 3 arctan (x)

= ln x^{2} + 1 + 3 arctan (x)

Add a constant to the solution

= ln x^{2} + 1 + 3 arctan (x) + C

x \to 4 - lim (16 - x^{2})

\frac{d y}{d x}, 1 = (\frac{y}{x})

f (x) = 8 cos (t)

x \to 16 lim (\frac{sin ( x )}{x})

x y^{^{'}} - 2 y = - 2 x^{2} + 2 x - 2

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