What is the integral of (cos(x)-sin(2x)) ?

The integral of (cos(x)-sin(2x)) is sin(x)+1/2 cos(2x)+C

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integral of (cos(x)-sin(2x))

\int (cos (x) - sin (2 x)) d x

sin (x) + \frac{1}{2} cos (2 x) + C

Solution steps

\int cos (x) - sin (2 x) 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 cos (x) d x - \int sin (2 x) d x

\int cos (x) d x = sin (x)

\int sin (2 x) d x = - \frac{1}{2} cos (2 x)

= sin (x) - (- \frac{1}{2} cos (2 x))

Simplify

= sin (x) + \frac{1}{2} cos (2 x)

Add a constant to the solution

= sin (x) + \frac{1}{2} cos (2 x) + C

\frac{d y}{d x} + 2 y = 35 sin (x) + 15 cos (x), y (0) = 9

\frac{\partial}{\partial y} (e^{- x} sec (y))

\frac{d}{d x} (x^{4} + x)

\frac{d}{d x} (sin^{2} (2 x))

What is the integral of (cos(x)-sin(2x)) ?
The integral of (cos(x)-sin(2x)) is sin(x)+1/2 cos(2x)+C