What is the integral of cos((mpix)/2) ?

The integral of cos((mpix)/2) is 2/(pim)sin((pimx)/2)+C

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integral of cos((mpix)/2)

\int cos (\frac{mπ x}{2}) d x

\frac{2}{πm} sin (\frac{πm x}{2}) + C

Solution steps

\int cos (\frac{mπ x}{2}) d x

Apply u-substitution

: \int \frac{cos ( \frac{u}{2} )}{πm} d u

= \int \frac{cos ( \frac{u}{2} )}{πm} d u

Take the constant out:

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

= \frac{1}{πm} \cdot \int cos (\frac{u}{2}) d u

Apply u-substitution

: \int cos (v) \cdot 2 d v

= \frac{1}{πm} \cdot \int cos (v) \cdot 2 d v

Take the constant out:

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

= \frac{1}{πm} \cdot 2 \cdot \int cos (v) d v

Use the common integral:

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

= \frac{1}{πm} \cdot 2 sin (v)

Substitute back

= \frac{1}{πm} \cdot 2 sin (\frac{m x π}{2})

Simplify

\frac{1}{πm} \cdot 2 sin (\frac{m x π}{2}) : \frac{2}{πm} sin (\frac{πm x}{2})

= \frac{2}{πm} sin (\frac{πm x}{2})

Add a constant to the solution

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

x \to \infty lim (1 - 2 x)

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

x (sin (x))

\frac{d}{d x} (- 1 + 4 sin (x))

\frac{\partial}{\partial x} (2)

What is the integral of cos((mpix)/2) ?
The integral of cos((mpix)/2) is 2/(pim)sin((pimx)/2)+C