What is the integral of xarctan(16x) ?

The integral of xarctan(16x) is 1/2 x^2arctan(16x)-1/512 (-arctan(16x)+16x)+C

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integral of xarctan(16x)

\int x arctan (16 x) d x

\frac{1}{2} x^{2} arctan (16 x) - \frac{1}{512} (- arctan (16 x) + 16 x) + C

Solution steps

\int x arctan (16 x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} arctan (16 x) - \int \frac{8 x ^{2}}{256 x ^{2} + 1} d x

= \frac{1}{2} x^{2} arctan (16 x) - \int \frac{8 x ^{2}}{256 x ^{2} + 1} d x

\int \frac{8 x ^{2}}{256 x ^{2} + 1} d x = \frac{1}{512} (- arctan (16 x) + 16 x)

= \frac{1}{2} x^{2} arctan (16 x) - \frac{1}{512} (- arctan (16 x) + 16 x)

Add a constant to the solution

= \frac{1}{2} x^{2} arctan (16 x) - \frac{1}{512} (- arctan (16 x) + 16 x) + C

\frac{\partial}{\partial y} (x + \frac{x}{x ^{2} + y ^{2}})

\frac{d y}{d x} = 4 e^{- 3 x}

\int e^{x} \cdot sin (e^{x}) d x

x \to 3 - lim (- (3)^{2} + 5 (3) - 5)

8 x - 2

What is the integral of xarctan(16x) ?
The integral of xarctan(16x) is 1/2 x^2arctan(16x)-1/512 (-arctan(16x)+16x)+C