What is the integral of pitan(pix) ?

The integral of pitan(pix) is -ln|cos(pix)|+C

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integral of pitan(pix)

\int π tan (π x) d x

- ln ∣ cos (π x) ∣ + C

Solution steps

\int π 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 tan (π x) d x

Apply u-substitution

: \int tan (u) \frac{1}{π} d u

= π \cdot \int 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 tan (u) d u

Rewrite using trig identities

= π \frac{1}{π} \cdot \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

: \int - \frac{1}{v} d v

= π \frac{1}{π} \cdot \int - \frac{1}{v} d v

Take the constant out:

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

= π \frac{1}{π} (- \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= π \frac{1}{π} (- ln ∣ v ∣)

Substitute back

= π \frac{1}{π} (- ln ∣ cos (π x) ∣)

Simplify

π \frac{1}{π} (- ln ∣ cos (π x) ∣) : - ln ∣ cos (π x) ∣

= - ln ∣ cos (π x) ∣

Add a constant to the solution

= - ln ∣ cos (π x) ∣ + C

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What is the integral of pitan(pix) ?
The integral of pitan(pix) is -ln|cos(pix)|+C