What is the integral of (27arctan(x))/(x^2) ?

The integral of (27arctan(x))/(x^2) is 27(-(arctan(x))/x+ln|x|-1/2 ln|x^2+1|)+C

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integral of (27arctan(x))/(x^2)

\int \frac{27 arctan ( x )}{x ^{2}} d x

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

Solution steps

\int \frac{27 arctan ( x )}{x ^{2}} d x

Take the constant out:

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

= 27 \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

= 27 (- \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

= 27 (- \frac{arctan ( x )}{x} - (- ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 1))

Simplify

= 27 (- \frac{arctan ( x )}{x} + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Add a constant to the solution

= 27 (- \frac{arctan ( x )}{x} + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

\frac{\partial}{\partial u} (\frac{u}{4 ( v - w ) ^{\frac{3}{2}}})

\frac{d y}{d x} = 1 + 5 y

7 + 4 x^{2} - 2 x^{3}

x \to 1 - lim (e^{x})

\frac{x + 6}{x + 2}

What is the integral of (27arctan(x))/(x^2) ?
The integral of (27arctan(x))/(x^2) is 27(-(arctan(x))/x+ln|x|-1/2 ln|x^2+1|)+C