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

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

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

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

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

Solution steps

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

Expand

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

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

\int \frac{5}{x ( x ^{2} + 1 )} d x = 5 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

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

Add a constant to the solution

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

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