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

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

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

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

\frac{3}{2} ln x^{2} + 1 + 5 arctan (x) + C

Solution steps

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

Expand

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

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

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

= \frac{3}{2} ln x^{2} + 1 + 5 arctan (x)

Add a constant to the solution

= \frac{3}{2} ln x^{2} + 1 + 5 arctan (x) + C

\int \frac{3 x ^{2} - 6 x - 4}{x ^{3} - 3 x ^{2} - 4 x + 12} d x

\frac{d}{d x} (x sinh (x))

x \to 2 + lim (\frac{x ^{3} - 8}{x - 2})

\int arccot (11 y) d y

f (x) = arctan (2 x)

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