What is the integral of xarctan(8x) ?

The integral of xarctan(8x) is 1/2 x^2arctan(8x)-1/128 (-arctan(8x)+8x)+C

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integral of xarctan(8x)

\int x arctan (8 x) d x

\frac{1}{2} x^{2} arctan (8 x) - \frac{1}{128} (- arctan (8 x) + 8 x) + C

Solution steps

\int x arctan (8 x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} arctan (8 x) - \int \frac{4 x ^{2}}{64 x ^{2} + 1} d x

= \frac{1}{2} x^{2} arctan (8 x) - \int \frac{4 x ^{2}}{64 x ^{2} + 1} d x

\int \frac{4 x ^{2}}{64 x ^{2} + 1} d x = \frac{1}{128} (- arctan (8 x) + 8 x)

= \frac{1}{2} x^{2} arctan (8 x) - \frac{1}{128} (- arctan (8 x) + 8 x)

Add a constant to the solution

= \frac{1}{2} x^{2} arctan (8 x) - \frac{1}{128} (- arctan (8 x) + 8 x) + C

f (x) = cos^{2} (4 x)

\frac{d}{d x} ((6 x + e^{- x^{2}})^{5})

x \to 0 lim (\frac{2 x}{sin ( x )})

\int \frac{60 x ^{2} + 20 x}{10 x ^{3} + 5 x ^{2} + 3} d x

y = x x

What is the integral of xarctan(8x) ?
The integral of xarctan(8x) is 1/2 x^2arctan(8x)-1/128 (-arctan(8x)+8x)+C