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

The integral of 2xcos(x/3) is 2(3xsin(x/3)+9cos(x/3))+C

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integral of 2xcos(x/3)

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

2 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3})) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

= 2 (3 x sin (\frac{x}{3}) - \int 3 sin (\frac{x}{3}) d x)

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

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

Simplify

= 2 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3}))

Add a constant to the solution

= 2 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3})) + C

What is the integral of 2xcos(x/3) ?
The integral of 2xcos(x/3) is 2(3xsin(x/3)+9cos(x/3))+C