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

The integral of (arctan(2x))/(x^2) is -(arctan(2x))/x+2ln(2|x|)-ln|4x^2+1|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

- \frac{arctan ( 2 x )}{x} + 2 ln (2 ∣ x ∣) - ln 4 x^{2} + 1 + C

Solution steps

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

Apply u-substitution

: \int \frac{2 arctan ( u )}{u ^{2}} d u

= \int \frac{2 arctan ( u )}{u ^{2}} d u

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = 2 x

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

Simplify

2 (- \frac{arctan ( 2 x )}{2 x} - (- ln ∣ 2 x ∣ + \frac{1}{2} ln (2 x)^{2} + 1)) : - \frac{arctan ( 2 x )}{x} + 2 ln (2 ∣ x ∣) - ln 4 x^{2} + 1

= - \frac{arctan ( 2 x )}{x} + 2 ln (2 ∣ x ∣) - ln 4 x^{2} + 1

Add a constant to the solution

= - \frac{arctan ( 2 x )}{x} + 2 ln (2 ∣ x ∣) - ln 4 x^{2} + 1 + C

\int (9 - 2 x)^{11} d x

\int \frac{2 x + 4}{1 + ( x + 2 ) ^{2}} d x

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

\int \frac{x + 3}{x - 3} d x

\int (x^{3} - 3 x^{2} + 6 x - 20) d x

What is the integral of (arctan(2x))/(x^2) ?
The integral of (arctan(2x))/(x^2) is -(arctan(2x))/x+2ln(2|x|)-ln|4x^2+1|+C