What is the integral of (4x+1)cos(x/3) ?

The integral of (4x+1)cos(x/3) is 4(3xsin(x/3)+9cos(x/3))+3sin(x/3)+C

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integral of (4x+1)cos(x/3)

\int (4 x + 1) cos (\frac{x}{3}) d x

4 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3})) + 3 sin (\frac{x}{3}) + C

Solution steps

\int (4 x + 1) cos (\frac{x}{3}) d x

Expand

(4 x + 1) cos (\frac{x}{3}) : 4 x cos (\frac{x}{3}) + cos (\frac{x}{3})

= \int 4 x cos (\frac{x}{3}) + 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 4 x cos (\frac{x}{3}) d x + \int cos (\frac{x}{3}) d x

\int 4 x cos (\frac{x}{3}) d x = 4 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3}))

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

= 4 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3})) + 3 sin (\frac{x}{3})

Add a constant to the solution

= 4 (3 x sin (\frac{x}{3}) + 9 cos (\frac{x}{3})) + 3 sin (\frac{x}{3}) + C

What is the integral of (4x+1)cos(x/3) ?
The integral of (4x+1)cos(x/3) is 4(3xsin(x/3)+9cos(x/3))+3sin(x/3)+C