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

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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

\int \frac{1}{x 1 - x} d x

\int \frac{1}{1 + x ^{4}} d x

\int_{0}^{2} x^{5} e^{x^{3}} d x

4 x^{2} - 3, - x^{2} + 9

\frac{\partial}{\partial x} (x^{2} + 2 x y + 2 y^{2} - 4 x + 4 y)

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