What is the integral of arctan(5t) ?

The integral of arctan(5t) is tarctan(5t)-1/10 ln|25t^2+1|+C

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integral of arctan(5t)

\int arctan (5 t) d t

t arctan (5 t) - \frac{1}{10} ln 25 t^{2} + 1 + C

Solution steps

\int arctan (5 t) d t

Apply Integration By Parts

: t arctan (5 t) - \int \frac{5 t}{25 t ^{2} + 1} d t

= t arctan (5 t) - \int \frac{5 t}{25 t ^{2} + 1} d t

\int \frac{5 t}{25 t ^{2} + 1} d t = \frac{1}{10} ln 25 t^{2} + 1

= t arctan (5 t) - \frac{1}{10} ln 25 t^{2} + 1

Add a constant to the solution

= t arctan (5 t) - \frac{1}{10} ln 25 t^{2} + 1 + C

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

y = \frac{1}{x ( 8 + ln ( x ) )}

x y^{^{'}} + y = - 3 x y^{2}

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

(5 x + 2) + (5 y - 5) y^{^{'}} = 0

What is the integral of arctan(5t) ?
The integral of arctan(5t) is tarctan(5t)-1/10 ln|25t^2+1|+C