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

The integral of 2cos((pix)/2) is 4/pi sin((pix)/2)+C

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integral of 2cos((pix)/2)

\int 2 cos (\frac{π x}{2}) d x

\frac{4}{π} sin (\frac{π x}{2}) + C

Solution steps

\int 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

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

Apply u-substitution

: \int cos (u) \frac{2}{π} d u

= 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

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

Use the common integral:

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

= 2 \cdot \frac{2}{π} sin (u)

Substitute back

u = \frac{π x}{2}

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

Simplify

2 \cdot \frac{2}{π} sin (\frac{π x}{2}) : \frac{4}{π} sin (\frac{π x}{2})

= \frac{4}{π} sin (\frac{π x}{2})

Add a constant to the solution

= \frac{4}{π} sin (\frac{π x}{2}) + C

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What is the integral of 2cos((pix)/2) ?
The integral of 2cos((pix)/2) is 4/pi sin((pix)/2)+C