What is the integral from 0 to pi/6 of xsin(x+y) ?

The integral from 0 to pi/6 of xsin(x+y) is -pi/6 cos(y+pi/6)+sin(y+pi/6)-sin(y)

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integral from 0 to pi/6 of xsin(x+y)

\int_{0}^{\frac{π}{6}} x sin (x + y) d x

- \frac{π}{6} cos (y + \frac{π}{6}) + sin (y + \frac{π}{6}) - sin (y)

Solution steps

\int_{0}^{\frac{π}{6}} x sin (x + y) d x

Apply Integration By Parts

: [- x cos (y + x) - \int - cos (y + x) d x]_{0}^{\frac{π}{6}}

= [- x cos (y + x) - \int - cos (y + x) d x]_{0}^{\frac{π}{6}}

\int - cos (y + x) d x = - sin (y + x)

= [- x cos (y + x) - (- sin (y + x))]_{0}^{\frac{π}{6}}

Simplify

= [- x cos (y + x) + sin (y + x)]_{0}^{\frac{π}{6}}

Compute the boundaries:

- \frac{π}{6} cos (y + \frac{π}{6}) + sin (y + \frac{π}{6}) - sin (y)

= - \frac{π}{6} cos (y + \frac{π}{6}) + sin (y + \frac{π}{6}) - sin (y)

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What is the integral from 0 to pi/6 of xsin(x+y) ?
The integral from 0 to pi/6 of xsin(x+y) is -pi/6 cos(y+pi/6)+sin(y+pi/6)-sin(y)