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

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

Popular >

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

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

\frac{d}{d x} (2 (x + 4)^{\frac{3}{2}})

n = 1 \sum \infty (1 - \frac{5}{n})^{n}

\frac{4}{3} π x^{3}

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