What is the integral of pisec(pix)tan(pix) ?

The integral of pisec(pix)tan(pix) is sec(pix)+C

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integral of pisec(pix)tan(pix)

\int π sec (π x) tan (π x) d x

sec (π x) + C

Solution steps

\int π sec (π x) tan (π x) d x

Take the constant out:

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

= π \cdot \int sec (π x) tan (π x) d x

Apply u-substitution

: \int sec (u) tan (u) \frac{1}{π} d u

= π \cdot \int sec (u) tan (u) \frac{1}{π} d u

Take the constant out:

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

= π \frac{1}{π} \cdot \int sec (u) tan (u) d u

Apply u-substitution

: \int 1 d v

= π \frac{1}{π} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= π \frac{1}{π} \cdot 1 \cdot v

Substitute back

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

Simplify

π \frac{1}{π} \cdot 1 \cdot sec (π x) : sec (π x)

= sec (π x)

Add a constant to the solution

= sec (π x) + C

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What is the integral of pisec(pix)tan(pix) ?
The integral of pisec(pix)tan(pix) is sec(pix)+C