What is the integral of sec(x/4) ?

The integral of sec(x/4) is 4ln|tan(x/4)+sec(x/4)|+C

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integral of sec(x/4)

\int sec (\frac{x}{4}) d x

4 ln tan (\frac{x}{4}) + sec (\frac{x}{4}) + C

Solution steps

\int sec (\frac{x}{4}) d x

Apply u-substitution

: \int sec (u) \cdot 4 d u

= \int sec (u) \cdot 4 d u

Take the constant out:

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

= 4 \cdot \int sec (u) d u

Use the common integral:

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

= 4 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = \frac{x}{4}

= 4 ln tan (\frac{x}{4}) + sec (\frac{x}{4})

Add a constant to the solution

= 4 ln tan (\frac{x}{4}) + sec (\frac{x}{4}) + C

y^{^{''}} + 9 y = 2 sin (x) + cos (x)

\frac{d y}{d x} - x y = x

25 y^{^{''}} + 90 y^{^{'}} + 81 y = 0

\frac{d}{d x} (65611 2^{x})

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

What is the integral of sec(x/4) ?
The integral of sec(x/4) is 4ln|tan(x/4)+sec(x/4)|+C