What is the integral of sec(1/2 x)tan(1/2 x) ?

The integral of sec(1/2 x)tan(1/2 x) is 2sec(x/2)+C

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integral of sec(1/2 x)tan(1/2 x)

\int sec (\frac{1}{2} x) tan (\frac{1}{2} x) d x

2 sec (\frac{x}{2}) + C

Solution steps

\int sec (\frac{1}{2} x) tan (\frac{1}{2} x) d x

Rewrite using trig identities

= \int \frac{sin ( \frac{x}{2} )}{cos ^{2} ( \frac{x}{2} )} d x

Rewrite using trig identities

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

Apply u-substitution

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

Apply u-substitution

= 2 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 2 \cdot 1 \cdot v

Substitute back

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

Simplify

= 2 sec (\frac{x}{2})

Add a constant to the solution

= 2 sec (\frac{x}{2}) + C

What is the integral of sec(1/2 x)tan(1/2 x) ?
The integral of sec(1/2 x)tan(1/2 x) is 2sec(x/2)+C