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

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

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integral of sin(x/2)

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

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

Solution steps

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

Apply u-substitution

: \int sin (u) \cdot 2 d u

= \int sin (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 2 (- cos (u))

Substitute back

u = \frac{x}{2}

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

Simplify

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

Add a constant to the solution

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

f (x) = \frac{1}{cos ( x )}

\frac{d}{d y} (2 x y^{4} e^{y} + 2 x y^{3} + y)

y^{^{''}} + 4 y^{^{'}} + 6 y = cos (x), y (0) = 0, y^{^{'}} (0) = 2

n = 1 \sum \infty \frac{x ^{n}}{2 n - 1}

\frac{\partial}{\partial x} (e^{x z})