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

The integral of x*sin(k*x) is -1/k xcos(kx)+1/(k^2)sin(kx)+C

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integral of xsin(kx)

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

- \frac{1}{k} x cos (k x) + \frac{1}{k ^{2}} sin (k x) + C

Solution steps

\int x sin (k x) d x

Apply Integration By Parts

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

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

\int - \frac{1}{k} cos (k x) d x = - \frac{1}{k ^{2}} sin (k x)

= - \frac{1}{k} x cos (k x) - (- \frac{1}{k ^{2}} sin (k x))

Simplify

= - \frac{1}{k} x cos (k x) + \frac{1}{k ^{2}} sin (k x)

Add a constant to the solution

= - \frac{1}{k} x cos (k x) + \frac{1}{k ^{2}} sin (k x) + C

(4 x)^{6 x}

\frac{\partial}{\partial x} (e^{- y} cos (x))

\int \frac{1}{x + x + 1} d x

\frac{d y}{d t} = e^{- 3 t}

cos^{x} (x)

What is the integral of x*sin(k*x) ?
The integral of x*sin(k*x) is -1/k xcos(kx)+1/(k^2)sin(kx)+C