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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

Add a constant to the solution

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

l a pl a ce t r an s f or m t \cdot sin (k \cdot t)

x \to - 4 lim (- 5 x - 8)

x y^{^{'}} - 2 y = x^{2} - x - 2

y^{^{'}} = x e^{y - x^{2}}

a re a x = 2 y^{2}, x = 2

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