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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

\frac{\partial}{\partial y} (x y e^{- 1 y})

x \to 1 lim (sin (\frac{π x}{2}))

\int - 3 \cdot x + 2 x d x

\int \frac{x ^{2}}{x ^{2} - 4} d x

\frac{- 5}{x ^{5}}

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