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

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

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

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

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

Solution steps

\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

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

Simplify

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

Add a constant to the solution

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

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

\frac{d}{d x} (5 x^{3} ln (x))

\frac{d}{d x} (6 x - x^{2})

y^{^{'}} + y = e^{t} + 11

s^{2} + 7 s + 12

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