What is the integral of pi/2 cos((pix)/2) ?

The integral of pi/2 cos((pix)/2) is sin((pix)/2)+C

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integral of pi/2 cos((pix)/2)

\int \frac{π}{2} cos (\frac{π x}{2}) d x

sin (\frac{π x}{2}) + C

Solution steps

\int \frac{π}{2} cos (\frac{π x}{2}) d x

Take the constant out:

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

= \frac{π}{2} \cdot \int cos (\frac{π x}{2}) d x

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{π}{2} \cdot \frac{2}{π} sin (u)

Substitute back

u = \frac{π x}{2}

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

Simplify

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

= sin (\frac{π x}{2})

Add a constant to the solution

= sin (\frac{π x}{2}) + C

What is the integral of pi/2 cos((pix)/2) ?
The integral of pi/2 cos((pix)/2) is sin((pix)/2)+C