What is the integral of-2/7 xcos(2/3 x) ?

The integral of-2/7 xcos(2/3 x) is -3/14 (2xsin(2/3 x)+3cos(2/3 x))+C

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integral of-2/7 xcos(2/3 x)

\int - \frac{2}{7} x cos (\frac{2}{3} x) d x

- \frac{3}{14} (2 x sin (\frac{2}{3} x) + 3 cos (\frac{2}{3} x)) + C

Solution steps

\int - \frac{2}{7} x cos (\frac{2}{3} x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{2}{7} \cdot \int x cos (\frac{2}{3} x) d x

Apply u-substitution

: \int \frac{9 u cos ( u )}{4} d u

= - \frac{2}{7} \cdot \int \frac{9 u cos ( u )}{4} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{2}{7} \cdot \frac{9}{4} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= - \frac{2}{7} \cdot \frac{9}{4} (u sin (u) - \int sin (u) d u)

\int sin (u) d u = - cos (u)

= - \frac{2}{7} \cdot \frac{9}{4} (u sin (u) - (- cos (u)))

Substitute back

u = \frac{2}{3} x

= - \frac{2}{7} \cdot \frac{9}{4} (\frac{2}{3} x sin (\frac{2}{3} x) - (- cos (\frac{2}{3} x)))

Simplify

- \frac{2}{7} \cdot \frac{9}{4} (\frac{2}{3} x sin (\frac{2}{3} x) - (- cos (\frac{2}{3} x))) : - \frac{3}{14} (2 x sin (\frac{2}{3} x) + 3 cos (\frac{2}{3} x))

= - \frac{3}{14} (2 x sin (\frac{2}{3} x) + 3 cos (\frac{2}{3} x))

Add a constant to the solution

= - \frac{3}{14} (2 x sin (\frac{2}{3} x) + 3 cos (\frac{2}{3} x)) + C

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What is the integral of-2/7 xcos(2/3 x) ?
The integral of-2/7 xcos(2/3 x) is -3/14 (2xsin(2/3 x)+3cos(2/3 x))+C