What is the integral of ((arctan(5x)))/(1+25x^2) ?

The integral of ((arctan(5x)))/(1+25x^2) is 1/10 arctan^2(5x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of ((arctan(5x)))/(1+25x^2)

\int \frac{( arctan ( 5 x ) )}{1 + 25 x ^{2}} d x

\frac{1}{10} arctan^{2} (5 x) + C

Solution steps

\int \frac{arctan ( 5 x )}{1 + 25 x ^{2}} d x

Apply Integral Substitution

= \int \frac{arctan ( u )}{5 ( 1 + u ^{2} )} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{arctan ( u )}{1 + u ^{2}} d u

Apply u-substitution

= \frac{1}{5} \cdot \int v d v

Apply the Power Rule

= \frac{1}{5} \cdot \frac{v ^{2}}{2}

Substitute back

= \frac{1}{5} \cdot \frac{arctan ^{2} ( 5 x )}{2}

Simplify

\frac{1}{5} \cdot \frac{arctan ^{2} ( 5 x )}{2} : \frac{1}{10} arctan^{2} (5 x)

= \frac{1}{10} arctan^{2} (5 x)

Add a constant to the solution

= \frac{1}{10} arctan^{2} (5 x) + C

What is the integral of ((arctan(5x)))/(1+25x^2) ?
The integral of ((arctan(5x)))/(1+25x^2) is 1/10 arctan^2(5x)+C