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

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

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

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

ln ∣ x ∣ + 2 arctan (x) + C

Solution steps

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

Take the partial fraction of

\frac{( x + 1 ) ^{2}}{x ^{3} + x} : \frac{1}{x} + \frac{2}{x ^{2} + 1}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + 2 arctan (x)

Add a constant to the solution

= ln ∣ x ∣ + 2 arctan (x) + C

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

3 x^{\frac{1}{2}}

f (x) = 3 x cos (x), at x = π

\frac{\partial}{\partial x} (x^{2} + 3)

9 y^{^{''}} + 9 y = 0

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