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

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

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integral of (2arctan(x))/(x^3)

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

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

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{arctan ( x )}{x ^{3}} d x

Apply Integration By Parts

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

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

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

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

Simplify

2 (- \frac{arctan ( x )}{2 x ^{2}} - (- \frac{1}{2} (- \frac{1}{x} - arctan (x)))) : - \frac{arctan ( x )}{x ^{2}} - \frac{1}{x} - arctan (x)

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

Add a constant to the solution

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

y = \frac{2}{x}, (7, \frac{2}{7})

x \to 0 lim (x^{2} \frac{sin ( 1 )}{x})

cos (2 t) - sin (2 t)

x \to \frac{a}{2} lim (csc (x))

y = sin (7 x) + sin^{2} (7 x), (0, 0)

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