What is the integral of xsin(ax) ?

The integral of xsin(ax) is -1/a xcos(ax)+1/(a^2)sin(ax)+C

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integral of xsin(ax)

\int x sin (a x) d x

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

Solution steps

\int x sin (a x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int ln (cos (x)) d x

y^{^{'}} = 1 - \frac{y}{100}

sin (x)

5 (x^{2 n - 1}) \cdot (2 x^{3 n + 1})^{2}

f (x) = x e^{\frac{1}{x}}

What is the integral of xsin(ax) ?
The integral of xsin(ax) is -1/a xcos(ax)+1/(a^2)sin(ax)+C