What is the integral of 1/2 (x+1/2 sin(2x)) ?

The integral of 1/2 (x+1/2 sin(2x)) is 1/2 ((x^2)/2-1/4 cos(2x))+C

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integral of 1/2 (x+1/2 sin(2x))

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

\frac{1}{2} (\frac{x ^{2}}{2} - \frac{1}{4} cos (2 x)) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int x + \frac{1}{2} 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

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

\int x d x = \frac{x ^{2}}{2}

\int \frac{1}{2} sin (2 x) d x = - \frac{1}{4} cos (2 x)

= \frac{1}{2} (\frac{x ^{2}}{2} - \frac{1}{4} cos (2 x))

Add a constant to the solution

= \frac{1}{2} (\frac{x ^{2}}{2} - \frac{1}{4} cos (2 x)) + C

x \to 1 lim (x^{2} - 2 x)

\frac{\partial}{\partial y} (x^{2} - 5 x y + 3 y^{2})

x \to - 2 lim ((2 x - 1)^{2})

\int 4 x^{2} - 16 x + 16 d x

x \to 1 lim (\frac{x ^{4} - 1}{x - 1})

What is the integral of 1/2 (x+1/2 sin(2x)) ?
The integral of 1/2 (x+1/2 sin(2x)) is 1/2 ((x^2)/2-1/4 cos(2x))+C