What is the integral of (8x+1)cos(x/4) ?

The integral of (8x+1)cos(x/4) is 8(4xsin(x/4)+16cos(x/4))+4sin(x/4)+C

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integral of (8x+1)cos(x/4)

\int (8 x + 1) cos (\frac{x}{4}) d x

8 (4 x sin (\frac{x}{4}) + 16 cos (\frac{x}{4})) + 4 sin (\frac{x}{4}) + C

Solution steps

\int (8 x + 1) cos (\frac{x}{4}) d x

Expand

(8 x + 1) cos (\frac{x}{4}) : 8 x cos (\frac{x}{4}) + cos (\frac{x}{4})

= \int 8 x cos (\frac{x}{4}) + cos (\frac{x}{4}) 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 8 x cos (\frac{x}{4}) d x + \int cos (\frac{x}{4}) d x

\int 8 x cos (\frac{x}{4}) d x = 8 (4 x sin (\frac{x}{4}) + 16 cos (\frac{x}{4}))

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

= 8 (4 x sin (\frac{x}{4}) + 16 cos (\frac{x}{4})) + 4 sin (\frac{x}{4})

Add a constant to the solution

= 8 (4 x sin (\frac{x}{4}) + 16 cos (\frac{x}{4})) + 4 sin (\frac{x}{4}) + C

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What is the integral of (8x+1)cos(x/4) ?
The integral of (8x+1)cos(x/4) is 8(4xsin(x/4)+16cos(x/4))+4sin(x/4)+C