What is the integral of xsin(x/3) ?

The integral of xsin(x/3) is -3xcos(x/3)+9sin(x/3)+C

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integral of xsin(x/3)

\int x sin (\frac{x}{3}) d x

- 3 x cos (\frac{x}{3}) + 9 sin (\frac{x}{3}) + C

Solution steps

\int x sin (\frac{x}{3}) d x

Apply Integration By Parts

: - 3 x cos (\frac{x}{3}) - \int - 3 cos (\frac{x}{3}) d x

= - 3 x cos (\frac{x}{3}) - \int - 3 cos (\frac{x}{3}) d x

\int - 3 cos (\frac{x}{3}) d x = - 9 sin (\frac{x}{3})

= - 3 x cos (\frac{x}{3}) - (- 9 sin (\frac{x}{3}))

Simplify

= - 3 x cos (\frac{x}{3}) + 9 sin (\frac{x}{3})

Add a constant to the solution

= - 3 x cos (\frac{x}{3}) + 9 sin (\frac{x}{3}) + C

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What is the integral of xsin(x/3) ?
The integral of xsin(x/3) is -3xcos(x/3)+9sin(x/3)+C