What is the integral of sec(t/2) ?

The integral of sec(t/2) is 2ln|tan(t/2)+sec(t/2)|+C

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integral of sec(t/2)

\int sec (\frac{t}{2}) d t

2 ln tan (\frac{t}{2}) + sec (\frac{t}{2}) + C

Solution steps

\int sec (\frac{t}{2}) d t

Rewrite using trig identities

= \int \frac{1}{cos ( \frac{t}{2} )} d t

Apply u-substitution

: \int 2 sec (u) d u

= \int 2 sec (u) d u

Take the constant out:

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

= 2 \cdot \int sec (u) d u

Use the common integral:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= 2 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = \frac{t}{2}

= 2 ln tan (\frac{t}{2}) + sec (\frac{t}{2})

Add a constant to the solution

= 2 ln tan (\frac{t}{2}) + sec (\frac{t}{2}) + C

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What is the integral of sec(t/2) ?
The integral of sec(t/2) is 2ln|tan(t/2)+sec(t/2)|+C