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

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

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integral of 1/(cos(x/2))

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

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

Solution steps

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

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{x}{2}

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

Add a constant to the solution

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

x \to 3 + lim (\frac{x + 4}{x + 3})

\int e^{x} + 3 x - x^{2} d x

\frac{1}{3 x ^{5}}

\int_{1}^{2} (6 x - 1) d x

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