What is the integral of (5x+2)cos(x/3) ?

The integral of (5x+2)cos(x/3) is 5(3xsin(x/3)+9cos(x/3))+6sin(x/3)+C

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integral of (5x+2)cos(x/3)

\int (5 x + 2) cos (\frac{x}{3}) d x

5 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3})) + 6 sin (\frac{x}{3}) + C

Solution steps

\int (5 x + 2) cos (\frac{x}{3}) d x

Expand

(5 x + 2) cos (\frac{x}{3}) : 5 x cos (\frac{x}{3}) + 2 cos (\frac{x}{3})

= \int 5 x cos (\frac{x}{3}) + 2 cos (\frac{x}{3}) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 5 x cos (\frac{x}{3}) d x + \int 2 cos (\frac{x}{3}) d x

\int 5 x cos (\frac{x}{3}) d x = 5 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3}))

\int 2 cos (\frac{x}{3}) d x = 6 sin (\frac{x}{3})

= 5 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3})) + 6 sin (\frac{x}{3})

Add a constant to the solution

= 5 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3})) + 6 sin (\frac{x}{3}) + C

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What is the integral of (5x+2)cos(x/3) ?
The integral of (5x+2)cos(x/3) is 5(3xsin(x/3)+9cos(x/3))+6sin(x/3)+C