What is the integral of (3x+5)cos(x/4) ?

The integral of (3x+5)cos(x/4) is 4sin(x/4)(3x+5)+48cos(x/4)+C

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integral of (3x+5)cos(x/4)

\int (3 x + 5) cos (\frac{x}{4}) d x

4 sin (\frac{x}{4}) (3 x + 5) + 48 cos (\frac{x}{4}) + C

Solution steps

\int (3 x + 5) cos (\frac{x}{4}) d x

Apply Integration By Parts

= 4 sin (\frac{x}{4}) (3 x + 5) - \int 12 sin (\frac{x}{4}) d x

\int 12 sin (\frac{x}{4}) d x = - 48 cos (\frac{x}{4})

= 4 sin (\frac{x}{4}) (3 x + 5) - (- 48 cos (\frac{x}{4}))

Simplify

= 4 sin (\frac{x}{4}) (3 x + 5) + 48 cos (\frac{x}{4})

Add a constant to the solution

= 4 sin (\frac{x}{4}) (3 x + 5) + 48 cos (\frac{x}{4}) + C

What is the integral of (3x+5)cos(x/4) ?
The integral of (3x+5)cos(x/4) is 4sin(x/4)(3x+5)+48cos(x/4)+C