What is the integral of sec((pix)/2)tan((pix)/2) ?

The integral of sec((pix)/2)tan((pix)/2) is 2/pi sec((pix)/2)+C

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integral of sec((pix)/2)tan((pix)/2)

\int sec (\frac{π x}{2}) tan (\frac{π x}{2}) d x

\frac{2}{π} sec (\frac{π x}{2}) + C

Solution steps

\int sec (\frac{π x}{2}) tan (\frac{π x}{2}) d x

Apply u-substitution

= \int sec (u) tan (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{2}{π} \cdot \int sec (u) tan (u) d u

Apply u-substitution

= \frac{2}{π} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{2}{π} \cdot 1 \cdot v

Substitute back

= \frac{2}{π} \cdot 1 \cdot sec (\frac{π x}{2})

Simplify

= \frac{2}{π} sec (\frac{π x}{2})

Add a constant to the solution

= \frac{2}{π} sec (\frac{π x}{2}) + C

What is the integral of sec((pix)/2)tan((pix)/2) ?
The integral of sec((pix)/2)tan((pix)/2) is 2/pi sec((pix)/2)+C