What is the integral of xsin(kwx) ?

The integral of xsin(kwx) is -1/(wk)xcos(wkx)+1/(w^2k^2)sin(wkx)+C

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integral of xsin(kwx)

\int x sin (k w x) d x

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

Solution steps

\int x sin (k w x) d x

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

\int_{0}^{2 π} sin (x) sin (x + 1) d x

\int 2 v d v

f (x) = lo g_{4} (x)

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

t^{2} e^{4 t}

What is the integral of xsin(kwx) ?
The integral of xsin(kwx) is -1/(wk)xcos(wkx)+1/(w^2k^2)sin(wkx)+C