What is the integral of arctan(2y) ?

The integral of arctan(2y) is yarctan(2y)-1/4 ln|4y^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(2y)

\int arctan (2 y) d y

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

Solution steps

\int arctan (2 y) d y

Apply Integration By Parts

: y arctan (2 y) - \int \frac{2 y}{4 y ^{2} + 1} d y

= y arctan (2 y) - \int \frac{2 y}{4 y ^{2} + 1} d y

\int \frac{2 y}{4 y ^{2} + 1} d y = \frac{1}{4} ln 4 y^{2} + 1

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

Add a constant to the solution

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

y = ln (\frac{1 + e ^{x}}{1 - e ^{x}})

\frac{d y}{d x} = - \frac{y ^{2}}{1 + y ^{2}} e^{(- x)}

n = 1 \sum \infty \frac{2 n}{7 n - 2}

\int \frac{1}{1 + cos ( x )} d x

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

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