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

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

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integral of sec^2(1/2 x)

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

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

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

= \int 2 sec^{2} (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^{2} (u) d u

Use the common integral:

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

= 2 tan (u)

Substitute back

u = \frac{x}{2}

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

Add a constant to the solution

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

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