What is the integral of x*sin(4x) ?

The integral of x*sin(4x) is -1/4 xcos(4x)+1/16 sin(4x)+C

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integral of x*sin(4x)

\int x \cdot sin (4 x) d x

- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x) + C

Solution steps

\int x sin (4 x) d x

Apply Integration By Parts

: - \frac{1}{4} x cos (4 x) - \int - \frac{1}{4} cos (4 x) d x

= - \frac{1}{4} x cos (4 x) - \int - \frac{1}{4} cos (4 x) d x

\int - \frac{1}{4} cos (4 x) d x = - \frac{1}{16} sin (4 x)

= - \frac{1}{4} x cos (4 x) - (- \frac{1}{16} sin (4 x))

Simplify

= - \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)

Add a constant to the solution

= - \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x) + C

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

\int 12 e^{3 x - 1} d x

\int \frac{6}{( 1 - 2 x ) ^{3}} d x

x \frac{d y}{d x} + y = - 6 x y^{2}

x \to 2 lim (2 x^{2} - 4 x)

What is the integral of x*sin(4x) ?
The integral of x*sin(4x) is -1/4 xcos(4x)+1/16 sin(4x)+C