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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

= 3 (- \frac{arctan ( x )}{x} - (- ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 1))

Simplify

= 3 (- \frac{arctan ( x )}{x} + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Add a constant to the solution

= 3 (- \frac{arctan ( x )}{x} + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

\frac{\partial}{\partial x} ((e^{x} - 2 x) cos (y))

\frac{d y}{d x} + a y - b = 0

f (x) = 5 x^{2} - 4 x - 2, at x = 1

t y^{^{'}} + 10 y = \frac{1}{t}

\int 7 \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

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