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

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

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integral of 5/(x^3+x)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Take the partial fraction of

\frac{1}{x ^{3} + x} : \frac{1}{x} - \frac{x}{x ^{2} + 1}

= 5 \cdot \int \frac{1}{x} - \frac{x}{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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Add a constant to the solution

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

\int csc^{2} (3 t) d t

\int cos^{5} (2 x) sin^{2} (2 x) d x

\int - 2 x - 2 d x

\int \frac{2 sin ( x )}{5} d x

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